Air-cooling analysis of AZ31B magnesium alloy plate: Experimental verification, numerical simulation and mathematical modeling

Journal of Alloys and Compounds 695 (2017) 1838e1853

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Air-cooling analysis of AZ31B magnesium alloy plate: Experimental verification, numerical simulation and mathematical modeling Weitao Jia, Yan Tang, Qichi Le*, Jianzhong Cui Key Laboratory of Electromagnetic Processing of Materials, Ministry of Education, Northeastern University, Shenyang 110819, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 3 September 2016 Received in revised form 30 October 2016 Accepted 2 November 2016 Available online 3 November 2016

The air-cooling transport process of AZ31B magnesium alloy plate was investigated and carried out under different initial cooling temperatures and different sample sizes at a transport velocity of 0.3 m/s. In order to provide the theoretical basis for the actual industrial rolling production, a detailed analysis of the temperature distributed along both the width direction and the thickness direction under different conditions was performed. Through the processing of the experimental data, the emissivity of AZ31B plate in the air-cooling transport process, as an important heat transfer parameter, was modeled accurately. After that, based on the establishment of the emissivity calculation model, the empirical formula of Stefan-Boltzmann was modified and optimized to be the temperature control model for the air-cooling process in consideration of the heat transfer characteristics of magnesium alloy. Finally, combining the finite element numerical simulation and the experimental results, comprehensive heat transfer coefficient between the AZ31B plate and the external environment during the transport process was accurately defined and further fitted about the experimental parameters. © 2016 Elsevier B.V. All rights reserved.

Keywords: Transport process Rolling production Emissivity calculation Temperature control Heat transfer coefficient

1. Introduction Magnesium alloy is widely used for many components in the aerospace, transportation, chemicals, rockets and other industries due to their excellent properties, such as low density and high specific strength, etc. [1]. Because of the adaptive advantages of material and the technical superiorities such as high efficiency and high performance, the continuous and short process of hot rolling of magnesium alloy (see Fig. 1(a)) gradually becomes the most promising and potential application in the actual industrial production of medium Mg plate as well as Mg sheet. With a hexagonal close-packed lattice structure (hcp), magnesium alloy has a limited number of slip systems, especially at room temperature [2,3], resulting in poor formability and limited ductility at the temperature. But, the increase of deformation temperature can increase the number of slip systems and facilitate the activation of the non-basal slip, and then improve the forming performance of materials [4]. Jeong and Ha [5] found that AZ31 Mg alloy sheet could be successfully rolled at 200
C with various reductions of 18, 30 and 50% by the conventional rolling method, but only when the

* Corresponding author. E-mail addresses: [email protected], [email protected] (Q. Le). http://dx.doi.org/10.1016/j.jallcom.2016.11.017 0925-8388/© 2016 Elsevier B.V. All rights reserved.

temperature is higher than 300
C, cracks could be prevented. Thus, achieving the high precision control of temperature is an important link to realize the high efficiency rolling process of Mg alloy. Through the analysis of the preparation technology shown in Fig. 1(a), we can divide it into five processes: heating process, transport process (①/⑤), rolling process, finishing process and coiling process. As the main process, rolling process possesses complex heat transfer factors (the heat exchange between the plate and rolls, conduction and friction at the plate/work rolls interfaces, plastic deformation heat and conduction inside the plate, and convection and radiation to the environment air (see Fig. 1(b)), which is the research content that the existing researches focused on. Considering the influence of these factors, we also established the models of temperature distributed in the different deformation regions (backward slip zone and forward slip zone) by mathematical analysis methods [6]. Compared with the rolling process, temperature change of the plate in the transport process is much smaller, then in the process of production nowadays, the production experience was usually used to estimate the degree of temperature drop, and then the plate temperature in the deformation process can be ensured by means of increasing the preheating temperature. But for a wide range of plate specifications and deformation conditions, a rough estimate of the temperature drop during the process will inevitably lead to the deviation of the

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Fig. 1. (a) Flow chart for continuous and short process of hot rolling of magnesium alloy, (b) A schematic diagram of heat transfer factors in the rolling process.

accuracy of the pre-heating in the heating process. This will not be able to accurately achieve the design temperature at the end of the transport process: when the temperature is lower than the design value, plate needs to be shipped back to the heating furnace once again in order to make up the temperature; when the temperature is higher, the plate temperature will exceed the most appropriate deformation temperature and seriously affect the formability of the sheet, which is attributed to the intense sensibility of the Mg alloys forming properties on temperature [7]. In particular, the related researches on the transport process, another important process, in the rolling technology of AZ31B are very limited, and the quantitative study on the law of air cooling about AZ31B magnesium alloy plate have not yet been reported in detail. Therefore, the present work aims to reveal the cooling law of AZ31B plate in the transport process and perform the quantitative analysis on some important air cooling parameters in order to achieve the precise control of temperature after air cooling in the transport process of industrial production. 2. Experimental procedure The experimental material is as-cast AZ31B magnesium alloy blank with the sample size of 1000 mm  800 mm  300 mm and the nominal chemical composition (wt.%) is shown in Fig. 2(a). In order to remove the internal stress and make the eutectic phase in the magnesium alloy sheet dissolve into the matrix, the research blank was successively exposed to the so-called heat soaking period of 16 h at 400
C in an industrial furnace, obtaining the average grain size of 600 mm, as shown in Fig. 2 (b). But from the figure, the coexistence of equiaxial grains and coarse grains was clearly observed. Then, for avoiding the adverse influence of the nonuniform distribution of the grain structure, impurity, second

phase, etc. in the slab and increasing the comparability among the experimental data, specimens were taken from the interior of the blank at the center of the thickness with the same sampling method, and whose directions (length, width and thickness) were parallel to those of the sample blank, respectively. After that, specimens were polished by hand to form a cleaned metal surface, which can effectively avoid the adverse influence of the defect and the oxide covered on the surface. Then, specimens were successively exposed to the heating stage and so-called heat soaking period of 35 min at 250e400
C in an industrial furnace (a circulating air oven, in order to avoid the danger of hot spots on surface). And then they were brought out rapidly from the furnace and placed on the experimental installation which was designed for the air-cooling simulation of the plate in the transport process. To ensure sufficient time for the transport of the workpiece at about 0.3 m/s in a limited transmission distance, conveying rollers with forward and reverse drive were designed and used to realize the reciprocating transmission of workpiece (see Fig. 3, unit: mm). In the process and for measuring the temperature at different locations, u thermocouples (TC for short) made from exposed butt welding (i.e., the diameter of conductor was 0.06 mm) with a 2 mm outer diameter were used. TC 1, 2, and 3 were inserted inside the plate through holes (the diameter was 2 mm) drilled from the surface perpendicular to the center layer of the thickness, and then we filled them with magnesium alloy chips for covering the entrance of the hole [8]. TC 4, 5, and 6 were welded successfully on the surface. For 2 mm thin plate, due to the drill hole will seriously affect the accuracy of the measurement of the temperature, we only measure the temperature of the surface. Note that the detection of temperature data is based on the thermal cycle measurement theory, and two exposed conductor wires were joined together by butt welding in the study. If the degree of contact between two

Fig. 2. (a) Chemical composition of the experimental material (b) Internal microstructure of the blank.

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Fig. 3. Schematic diagram of temperature measurement.

conductor wires is not enough at the welding point, or short-circuit happens at other locations, the accuracy of temperature data collection will be greatly affected. In order to record the timevarying temperature data, a recorder (HIOKI LR8400-21) made in Japan with a high recording frequency (100 Hz) was utilized, which has been successfully applied in the tracking of temperature distributed in the thickness direction in the AZ31B plate hot rolling process by the members of our team [9]. Finally, the thermal curves were drawn and analyzed by means of the Origin software. 3. Experimental results and discussion In order to effectively reduce the re-heating that is needed to maintain the appropriate deformation temperature of the plate, the air-cooling transport process with short-distance transmission is generally designed and applied in the industrial rolling production of magnesium alloy sheet. Because of the short transmission distance, the temperature drop of the plate in the air-cooling process is not very large. Therefore, we only analyze the air cooling process ranging from the initial temperatures (250, 300, 350, and 400
C) to 100
C not 26
C (room-temperature) in the study. During the research, we designed an experimental scheme to track the temperature data of the plate along the thickness and the width direction (see Fig. 3). There is a big difference in the heat transfer mechanism at different locations of the plate, then temperature difference will inevitably exist inside the plate. In the study, the moment when the temperature of all measuring points is cooled to 100
C is considered to be the statistical end point of time. Therefore, experimental temperatureetime curves at different measuring positions were obtained. 3.1. Time-history curves correlated with temperature changes Take time-history curves of 8 mme300
C as an example, we can observe that all curves in Fig. 4 can be divided into three stages. Stage I is the heating stage. In the thickness direction, as the heat flux is transferred from the outside to the inside of the plate, the temperature rising speed of the surface is faster than that of the middle layer at the beginning. We take (TC 4 þ TC 5 þ TC 6)/3 as the average temperature of the surface layer (Ts) and (TC 1 þ TC 2 þ TC 3)/3 as the average temperature of the middle layer (TM). From Fig. 4, it can be observed that when the plate is heated for about 10 min, the temperature difference between the surface (Ts) and the middle layer (TM) reaches the maximum value 16.8
C. In the width direction, both the surface layer and the middle layer exist the phenomenon of high edge temperature and low center

Fig. 4. Time-history curves of temperature about 8 mme300
C.

temperature. The external surface of the plate undoubtedly is the direct acting part of the heating effect produced by the heat radiation of the heating furnace. And, if we divide the plate into 2i þ 1 unit elements along the width direction (see Fig. 5(a)), the heat exchange area of the edge element will be obviously larger than that of the central element, so that the edge element could absorb more heat. We take (TC 3 þ TC 6)/2 as the average temperature of the edge (Te) and (TC 1 þ TC 4)/2 as the average temperature of the center (Tc). Observed from Fig. 4, when the plate is heated for about 5 min, the difference between Te and Tc reaches the maximum value 7.2
C. As time continues, with the decrease of the surface heating rate and full thermal conduction inside the plate, the temperature difference in two directions begins to decrease. After being heated for about 30 min, the plate presents a relatively uniform temperature distribution. In the stage Ⅱ shown in Fig. 4, central temperature gradually tends to be the same as the surface temperature with the change of time, and the plate temperature overall is approximately equal to the ambient temperature in the furnace, which is no longer changed, and after soaking over a longer time, the temperature distribution is more uniform through the full heat conduction in the plate. Stage Ⅲ is air-cooling stage, which is the main research in this paper. In this stage, all the curves show a constant and equal initial temperature, indicating that temperature distribution of the plate is thermally uniform before the air-cooling stage and the

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Fig. 5. Schematic drawing of the plate after being divided for micro-units along the width direction: (a) in the heating process, (b) in the air-cooling process.

temperature is maintained at designed test one. Then, a sharp fall appears in the temperature of the plate for all the cases and later a small reduction during the same time interval in temperature is observed. This is due to the fact that in the early stage of air-cooling, radiation heat transfer is the main method for heat transfer. It is well-known that heat radiation is direct proportional to the fourth power of temperature according to the Stefan-Boltzmann's law (see Eq. (1)), then at this time curves show a sharp decline. However, in the later stage, the dominant effect of radiation heat transfer has been weakened, and natural-convection heat transfer gradually becomes more important. Based on Newton's law of cooling, heat convection is direct proportional to the first power of temperature (see Eq. (2)), therefore, the heat exchange rate between the plate and the environment decreased gently, and the temperature of the plate decreased slowly.


 4 dt dQr ¼ Aεs Ts4  Tex

(1)

dQc ¼ Ahc ðTs  Tex Þdt

(2)

Where Qr is heat radiation, Qc is heat convection, A is radiation surface area, ε is emissivity, s is a Stefan-Boltzmann constant, Ts is surface temperature of the plate, Tex is ambient temperature, and hc is convective heat transfer coefficient. In order to study the law of air cooling of AZ31B plate with different initial temperatures and thicknesses, the average temperature (T0, calculated by Eq. (3)) of magnesium plate was used for the temperature drop evaluation of the plate overall. The timehistory of the average temperature (T0 ) changes after ignoring the curves in the stage Ⅰ and Ⅱ was shown as Fig. 6.

8 ðTðTC4Þ þTðTC5Þ þTðTC6Þ Þ > > > h¼2mm < 3 T0¼ >ðTðTC1Þ þTðTC2Þ þTðTC3Þ þTðTC4Þ þTðTC5Þ þTðTC6Þ Þ > > : h¼8;15;31mm 6 (3) It is observed from Fig. 6 that the difference between the desired design initial temperature and the experimental value is small, the maximum residual value and the relative error is 6
C and 2%, respectively, at designed 300
C. It shows that the design scheme is successfully implemented, and the experimental results are extremely reliable. The temperature history of the plate has a strong dependence on the thickness and initial temperature when the length and width are constants. Under the same thickness condition, there are some differences between the air-cooling curves starting from the low initial temperature and those with the same temperature after a certain degree of cooling from the high initial temperature. This is caused by the uniformity of the

temperature in the plate. After a certain degree of cooling and compared with the initial cooling state, temperature difference in the plate will appear inevitably, which can be seen from the experimental curves at different locations. As everyone knows that the inhomogeneity of temperature distribution is an important factor to affect the rolling process of magnesium alloy then the subsequent research is focused on the temperature distribution law inside the plate along the thickness direction and the width direction. A comparative analysis of the air-cooling time (t), as an evaluation parameter of air cooling, was carried out under different test conditions. Fig. 6 shows that the time required to cool the AZ31B plate to 100
C is very different under different thicknesses compared with that under different initial temperatures. Under the same initial temperature conditions and when the length and width is kept constant at 300 and 200 mm, respectively, the greater the thickness is, the longer the time will be used for cooling the plate from the design temperature to 100
C. And, under the same conditions of the thickness, the length and the width, air-cooling time also has a strong positive correlation with the initial temperature. These are due to the fact that with the increase of the thickness or the initial temperature, the total heat transferred by the cooling from the design initial temperature to 100
C is increased, and hence the air cooling time will be increased. 3.2. Temperature distribution along the width direction Take 15 mm (250, 400
C) and 31 mm (250, 400
C) as examples, different air-cooling theories of AZ31B plate in the width direction on the change of initial temperature and thickness were studied. We used Tm and Tm' calculated by Eq. (4) as evaluation parameters to evaluate the uniformity of temperature distribution along the width direction.

8
> TðTC > > > > < Tm ¼
> > > TðTC > > 0 : Tm ¼

1Þ

1Þ

þ TðTC þ TðTC

 4Þ

 4Þ


 TðTC



2Þ

þ TðTC

5Þ

3Þ

þ TðTC

6Þ

2


 TðTC



(4)

2

Where Tm is the difference of temperature between the center (see Fig. 3, position 1 and 4) and the 1/4 (see Fig. 3, position 2 and 5) of the width, and Tm0 is that between the center and the edge (see Fig. 3, position 3 and 6) of the plate. Obviously, the larger their values, the worse the uniformity of temperature distribution will be. Time-history curves of Tm and Tm0 are shown as Fig. 7. All the curves presented in Fig. 7 (a) and (b) show approximate 0
C at starting point and ending point, indicating that the plate is thermally soaked inside the furnace and after air cooling, the temperature distribution is relatively uniform, respectively. From the figure, we can see that in the air cooling process of magnesium

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Fig. 6. Air-cooling curves of AZ31B plate under different thicknesses: (a) 2 mm, (b) 8 mm, (c) 15 mm, (d) 31 mm.

alloy plates, the temperature difference in the width exists and is very different, namely, uneven temperature distribution is apparent. This is because the workpiece itself is a conductor of heat, but its thermal conductivity is limited, so a certain degree of temperature gradient in the thickness direction and the width direction of the workpiece appears. In the process of air cooling, heat exchange with the external environment is mainly carried out by the radiation heat transfer and convection heat transfer. In the aircooling process, Mg plates were placed on the supporting rollers (see Fig. 3), therefore, in addition to the heat transfer between the plate and the ambient air and internal heat conduction, there is a part of heat dissipation, which is transmitted from the plate to the supporting rollers through the contact surface. However, it is a fact that the contact between the plate and the roller is a linear contact, leading to the heat transfer is much smaller than other forms of heat transfer, then in order to simplify the analysis, we ignored this part of the heat dissipation. Due to that the edge element has three heat radiating surfaces and the central element only has two radiating surfaces (see Fig. 5(b)), compared with the temperature of the center, edge temperature decreased rapidly, and according to the principle of heat transfer, it is a fact that the heat is always transferred from the position with the high temperature to the position with the low temperature, thus causing the temperature gradually decreases from the center to the edge of the plate. The extension trend of all curves in Fig. 7 is similar, and each curve has

an obvious peak point. We make statistics on related data of the peak points, including the peak value DTp, its emergence moment tp, as well as the average temperature of the plate at this moment Tap, as Table 1. By analyzing the data in the table we can get the following conclusions: 1) As the thickness increases under the same initial temperature condition and as the initial temperature increases under the same thickness, the peak value of the curves of both Tm and Tm' increases, which indicates that the worse the uniformity of the distribution will be under the above both circumstances. Through the above analysis, different heat exchange area with the external environment between the edges and the center of the plate is the main reason for the temperature gradient. When the thickness increases, the area of the side will increase, then the edge temperature drop will enhance, finally, the temperature gradient between the edge and the central part of the plate will intensify, resulting in the enhancement of inhomogeneity of temperature distribution. With the increasing of the initial temperature, the heat radiation effect gradually plays a key role, and according to the Stefan-Boltzmann's law, heat radiation is direct proportional to the fourth power of temperature (see Eq. (1)), therefore, the higher the temperature is, the greater the

Fig. 7. Time-history curves of Tm and Tm0 under different thicknesses and initial temperatures: (a) 250
C and (b) 400
C.

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Table 1 Statistics on related data of the peak points.

(31 mm 250
C)

Tm Tm0 Tm Tm0 Tm Tm0 Tm Tm0

(31 mm 400
C) (15 mm 250
C) (15 mm 400
C)

Peak value (
C)

Emergence moment of peak point (sec)

Average plate temperature (
C)

1.63 5.48 3.06 9.72 1.14 4.34 1.88 6.74

208 180 185 118 188 123 149 97

221 225 350 367 211 222 344 360

amount of thermal radiation per unit time will be, leading to that the temperature drop of the edge becomes more intensify. 2) Through the comparative analysis of the emergence moment of the peak point (tp), we found that compared with the DTp of the Tm (Center temperature minus that of 1/4 of the width), DTp of the Tm' (Center temperature minus that of edge of the width) appears earlier. This is because greater heat exchange with the environment at the edge is the direct cause of the formation of the temperature difference along the width, and then because of the thermal conductivity in the material itself, heat exchange will gradually affect the interior of the plate, which leads to the emergence moment of both parameters become not synchronized. Take the emergence moment of the peak point (tp) of the Tm' as the reference condition and make statistics of the DTp of Tm at the moment, we found that temperature difference between the 1/4 of the width and the center of the plate is less than 0.3 times the value that between the edge and the center. If temperature of plate has a linear decreasing trend from the center to the edge that should be 0.5 times, then we have reasons to believe that there is a sudden drop in temperature in the range of 1/4 of width distanced from edge, which is a key research area for the study of the temperature compensation. 3) Under different thickness conditions, for 250
C initial temperature, when the magnesium alloy plate is cooled by 25e36
C and for 400
C, when it is cooled by 33e56
C, inhomogeneous temperature distribution with the greatest degree will be generated. Then, in the process of AZ31B plate air-cooling, the temperature drop of the plate overall should not exceed the range in order to reduce the adverse effects of temperature difference in the width direction on the following process.

3.3. Temperature distribution along the thickness direction

(decreased (decreased (decreased (decreased (decreased (decreased (decreased (decreased

by by by by by by by by

30) 26) 54) 37) 36) 25) 56) 40)

soaked inside the furnace. Obviously, each curve has a peak point similar to that in Fig. 7 and presents a trend of first increasing and then decreasing. With the increase of the thickness of the plate, the cooling effect of the air in the air-cooling process will be prevented by more thermal resistance of the material, then it can not be quickly penetrated into the interior of the plate, which leads to a slowing down of the cooling rate in the central layer. To sum up, the value of Tcs increases with the increase of thickness. With the increase of the initial temperature, the Tcs also increases, which presents the same temperature distribution law as the distribution in the width direction and can be attributed to thermal radiation theory (the higher the temperature is, the greater the amount of thermal radiation per unit time will be). In the air cooling process, the temperature difference of AZ31B plate in the thickness direction is smaller than that in the width direction, which can be explained by an important non-dimensional parameter of heat transfer for qualitative analysis, Biot number (Bi, calculated by Eq. (6)) [10]. It shows the relative magnitude between the thermal resistance of the heat conduction and the thermal resistance of the surface heat transfer.

Bi ¼

hd

l

(6)

Where d is a half of the thickness value, h is the surface heat transfer coefficient, l is the coefficient of thermal conductivity of material. According to the material properties of conductor and surface heat transfer conditions, the effect of the change of the Biot number on the temperature distribution in the thickness direction can be summarized as the following three types (see Fig. 9). In the figure, t0 is the initial time of the air-cooling process, t∞ is the end time, and t is a sign of process. For the air-cooling process under different conditions, the value of Bi has been accurately calculated, 0.00008e0.00275, in section

In the thickness direction, because of the direct contact between the surface metal and the external environment, temperature drop phenomenon first appears in the surface layer. However, due to the segregation between the central layer of metal and the external environment, the air-cooling effect is affected by the thermal resistance of the material itself and can not be quickly penetrated into the interior of the plate, leading to the gradual decline of the temperature distribution from the center layer to the surface layer. Take 15 mm (250, 400
C) and 31 mm (250, 400
C) as examples again, we use Tcs as evaluation parameter to evaluate the uniformity of temperature distribution along the thickness, calculated by Eq. (5).


Tcs ¼

TðTC

1Þ

þ TðTC 3

2Þ

þ TðTC

 3Þ




TðTC

4Þ

þ TðTC

5Þ

þ TðTC

 6Þ

3 (5)

All the curves presented in Fig. 8 show approximate 0
C at starting point, which indicates again that the plate is thermally

Fig. 8. Time-history curves of Tcs under different thicknesses and initial temperatures.

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3.6.4. Apparently, temperature distribution of AZ31B plate in the thickness direction satisfies the TypeⅡ (see Fig. 9 (b)), which explains the phenomenon that the temperature distribution in the thickness is more uniform.

3.4. Quantitative analysis about cooling rate The curves of cooling rate changed with time can be obtained by first order differential treatment of all the curves in Fig. 6. The cooling rates calculated by Eq. (7) and presented in Fig. 10 have been estimated for the characterization of the temperature drop trend during the air-cooling process.

 !,    n X   ACR ¼ d Ti =n dt   

(7)

i¼1

Where Ti is the temperature measured by thermocouple i, n is the number of all the thermocouple placed in the plate, t is the aircooling time. Observed from Fig. 10, in the air-cooling process, cooling rates under different conditions are within the range of [0, 1.8] and have similar extension trends, proving that all curves belong to the cooling mechanism and cooling rate changes smoothly, proving the success of the experiment in a certain sense. It is found that the dependence of cooling rate on different cooling processes used to cool the hot AZ31B plates is very significant. Under different conditions, with the air cooling process continues, the cooling rate gradually decreases, and the smaller the thickness is, the severer the trend will be. The range value of the cooling rate under different conditions is shown in Table 2. For the cooling rates under different initial temperature and same thickness conditions, there is a great difference among them in the thickness of 2 mm. When the initial temperature is  300
C, the cooling rate is less than 1
C/s, and when >300
C, it is greater than 1
C/s. After a report given above, for the air-cooling process of the thin plate with a higher initial temperature (greater than 300
C), when the temperature difference between the plate and the environment changes, the cooling rate fluctuates greatly, so it will seriously affect the accuracy of the calculation of the plate temperature. When the initial temperature is increased from 300
C to 400
C, namely, the temperature difference between the plate and the environment gradually increases, the maximum cooling rate is increased from 0.865
C/s to 1.778
C/s. The above phenomenon shows that under this air-cooling condition, reducing the temperature difference between the plate and the environment will greatly reduce the temperature cooling rate of the plate then for the transport process of the AZ31B plate with thin thickness and

high initial temperature in industrial production, we should achieve the precise control of temperature by improving the way of thermal insulation to accurately predict the temperature at the end of the air cooling. However, for the thin plate with a lower initial temperature (less than 300
C), with the change of temperature difference, the cooling rate fluctuates relatively gently. So compared with improving thermal insulation, taking into account the temperature drop of the plate during the air cooling process in order to guide the preheating temperature of the plate in the heating stage, which can effectively realize the design temperature of the plate at the end of the air cooling and reduce the loss and input required by increasing the thermal insulation device in the transportation of the plate. For the plate within the range of 15e31 mm, with the increase of the initial temperature, the cooling rate increases, but there is no big difference in the value, and the maximum value is less than 0.692
C/s. It's easy to get that the effect of the temperature difference between the medium plate and the environment on the cooling rate is very small, which further illustrates that the temperature change is more stable during the air cooling of medium plate than that of the thin plate. Then, the design temperature of the plate after air cooling can also be realized by considering the temperature drop calculation and increasing the preheating temperature before the air cooling.

3.5. Establishment of temperature control model The above analysis described that heat radiation was direct proportional to the fourth power of temperature according to the Stefan-Boltzmann's law (see Eq. (1)) but direct proportional to the first power of temperature for heat convection based on Newton's law (see Eq. (2)). It is easy to draw a conclusion that in the air cooling process of the present study, radiation heat dissipation Qr is much larger than that of natural convection Qc and is the main heat transfer mechanism. Then we only consider the heat loss caused by radiation, but the other factors including the natural convection are included in the change of emissivity ε, namely, the emissivity is a variable. Finally, we can get the equation for calculating the heat loss Ql in a certain period of time t, and its differential form is

DQl zDQr ¼ εsA

"

Ts þ 273 100

4

 

Tex þ 273 100

4 # $Dt

(8)

In the equation, Tex is ambient temperature, approximate 26
C in the experiment, and A is heat exchange area, A ¼ 2(Bl þ BH þ lH), where B is the width, l is the length, and H is the thickness. Based on the empirical formula of heat transfer, the temperature drop caused by the heat loss is as

Fig. 9. Temperature distribution in the thickness direction with three types: (a) TypeⅠ, (b) TypeⅡ and (c) Type Ⅲ

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Fig. 10. Air-cooling rate under different thickness conditions: (a) 2 mm, (b) 8 mm, (c) 15 mm, and (d) 31 mm. Table 2 Cooling rates under different conditions (
C/s).

250 300 350 400




C
C
C
C

2 mm

8 mm

15 mm

31 mm

0.758e0.279 0.865e0.193 1.306e0.238 1.778e0.217

0.354e0.079 0.432e0.058 0.550e0.044 0.692e0.053

0.210e0.047 0.288e0.030 0.364e0.024 0.387e0.024

0.148e0.023 0.173e0.024 0.260e0.018 0.302e0.020

DQl ¼ BHlrcDT

(9)

Where r is density, and c is specific heat capacity. Bringing Eq. (9) into Eq. (8), we can get

8 εsA > > $f ðTs Þ,108 $Dt DT ¼ > < BHlrc   > > Ts þ 273 4 > : f ðTs Þ ¼  80 100

(10)

In the later stage, we have to deal with this equation for integral operation, but evidently, the processing of the key term f (Ts) in the equation will greatly increase the difficulty of the operation and cause the complexity of the formula. So f (Ts) needs to be simplified to the greatest extent. Because the traditional billet steel has a higher temperature (>1000
C) when it is transported on the track, then compared with Ts, the value of Tex is so small and usually ignored. However, the temperature in air-cooling process of magnesium alloy is much lower than that of traditional billet steel, then the effect of Tex can not be ignored. So we applied the mathematical method of equivalent transformation to deal with it under the plate temperature from 100 to 400
C. The equivalent treatment of the equation is carried out as Eq. (11). Using absolute relative error (ARE) to evaluate the simplified formula, which is calculated by Eq. (12).

f ðTs Þz

Where fi is the value of f (Ts), Fi is the calculated value by the simplified formula. Then, the correlation between the two equations is shown in Fig. 11. A better correlation (R2, 0.99996) between the calculation values of f (Ts) and values calculated by the simplified formula can be observed form Fig. 11. It can be seen that all of them are in the range of -6~6%, indicating that the simplified formula (see Eq. (11)) is appropriate for replacing the f (Ts) in Eq. (10). Through the explanation of the distribution of temperature along the width and thickness direction in section 3.2 and 3.3, three main assumptions that are reasonable aiming at the technological process are discussed in order to make a further simplification for Eq. (10). ① The surface temperature Ts is approximately equal to the average temperature of the plate Tm. ② According to the material characteristic, the density r of material under different temperatures is changed, then we take the mean value of them, namely, it has nothing to do with the temperature. ③ For specific heat capacity c and emissivity ε, we do the same simplification similar to the density. After simplification, temperature control model of AZ31B plate in air cooling process is as Eq. (13).

ZT1 T0

0:7364  0:002$T þ 1:7212  106 $T 2 dT ¼ T

Zt1 0

εsA  108 dt BHlrc (13)

Where T0 is the initial temperature before the air cooling, and T1 is

Ts ; T4½100
C; 400
C 0:73640:002$Ts þ1:7212106 $Ts2 (11)

ARE ¼

jfi  Fi j  100% fi

(12) Fig. 11. Comparison of calculation values before and after simplification.

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the temperature of the plate after a certain air-cooling time period of t1. Integral operation is performed on both sides of the equation. After integration, Eq. (13) is converted into

t¼

1.92e7.27% for the cooling process from different initial temperature to 100
C and 0.86e10.92% from that to 200
C. Obviously, the relative error ranges from about -10% to 10%, indicating that the established air-cooling control model of AZ31B plate is appropriate

i h
BHlrc 0:7364ðlnðT0 Þ  lnðT1 ÞÞ þ 0:002$ðT1  T0 Þ þ 0:8606  106 $ T02  T12 2εs  108 ðBl þ BH þ lHÞ

Thus, under a certain initial temperature conditions, through Eq. (14) we can easily calculate the average temperature of the AZ31B plate with different specifications after a certain air-cooling time period. For the determination of the emissivity ε, as an important parameter, D.R., Croft and D.G., Lilley [11] considered the emissivity of the steel plate as a relationship with the strip thickness (H): ε¼a,Hþb, where a and b are regression coefficients. When Eq. (14) was established, we considered other heat dissipation factors including the natural convection into the emissivity ε, and it is a truth that emissivity is related to the surface state and the surface temperature, then we considered the ε as a parameter related to the thickness and the initial temperature of the plate, ε¼ε(H, T0). On the basis of the experimental results, we used mathematical nonlinear fitting method to determine the form of the formula, ε¼ε(H, T0). Deterministic process is mainly divided into the following steps. First, all the constant values are brought into Eq. (14), including B ¼ 0.2 m, l ¼ 0.3 m, r ¼ 1745 kg/m3, c ¼ 1150 J/ (kg$
C), and s ¼ 5:67  108 W=ðm2 $
C 4 Þ. After that, the formula contains t, H, ε(H, T0), T1 and T0 only. Second, under different thickness H, statistics about the experimental cooling time t used for cooling the plate from different initial temperature to 100
C were carried out and taken into the equation. Then, the values of the ε(H, T0) corresponding to different conditions were determined and listed in Fig. 12. At last, all values of the ε(H, T0) were fitted with regard to the thickness and initial temperature by the mathematical nonlinear fitting method. As can be seen from Fig. 12, the emissivity has a very clear positive correlation on the thickness of the plate, especially presents an approximately linear relationship, then the calculation equation of the emissivity can be summarized as εðH; T0 Þ ¼ kðT0 Þ$H þ bðT0 Þ, where k(T0) and b(T0) is the slope and the intercept of the fitting line, respectively, which has something to do with the initial temperature T0. After that, we did fittings for the k(T0) and b(T0). Finally, we determined the theoretical calculation formula of the emissivity, as Eq. (15).

ε ¼ ð0:0112$T0 þ 7:9198Þ$H þ 0:0004$T0 þ 0:2984

for predicting the time needed for cooling the plate to the design temperature, then we can further control the air-cooling process of magnesium plate according to the theoretical equation. 3.6. Applications of the finite element method Through the basic equations of heat transfer (Eqs. (1) and (2)), it can be seen that the key problem to calculate the comprehensive heat transfer coefficient of the air cooling is how to solve the two variables accurately, including convection heat transfer coefficient and radiation heat transfer coefficient. The two variables and their changes with time are measured directly by experiments, which can solve the comprehensive heat transfer coefficient accurately, but as everyone knows the accurate measurement of surface temperature and heat flux is difficult to achieve. However, the numerical simulation method can provide the possibility and convenience for the determination of the comprehensive heat transfer coefficient, but the premise is how to ensure the accuracy of the calculation results of the temperature field. Therefore, we ensured the accuracy of the temperature field by comparing the measured temperature and simulated temperature, and then the boundary comprehensive heat transfer coefficient is obtained according to the result of temperature field simulation. After that, the solution for the comprehensive heat transfer coefficient can be transformed into a treatment process that the simulation results of the temperature field are compared with the experimental results to adjust the numerical simulation program so that the accuracy of the numerical simulation can meet the requirement. Flow chart for solving the comprehensive heat transfer coefficient under the condition of air cooling is shown as Fig. 15. Firstly, the temperature curve changed with time is obtained by

(15)

The non-linear fitting of emissivity model (see Eq. (15)) and calculation results through processing the experimental data above are shown in Fig. 13, from which we can see that Eq. (15) can accurately fit the emissivity about the thickness and initial aircooling temperature. Fitting accuracy of model is shown in Table 3, and the high fitting accuracy is further illustrated. Bring Eq. (15) into Eq. (14), we can get the accurate control model of AZ31B plate in the air cooling process. Through the model, we predicted the cooling time used for cooling the plate from different initial temperature to 200
C and 100
C, respectively, to verify the accuracy of the model. Experimental values compared with the values calculated by Eq. (14) are shown in Fig. 14. As shown in Fig. 14, the values of ARE calculated by Eq. (12) is

(14)

Fig. 12. ε(H, T0) corresponding to different conditions.

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Fig. 13. Comparison between calculation through processing experimental data and curved surface fitted. Table 3 Accuracy of emissivity fitting model. Parameters

Value

Sum of Residuals Average Residual Residual Sum of Squares (Absolute) Standard Error of the Estimate Coefficient of Multiple Determination (R^2) Multiple coefficient of determination (Ra^2)

1.257827175E11 7.861419848E13 1.9805037073E02 4.0625358535E02 0.968377741 0.956222176

experiment. Then, according to the literature references [12e15], we initially identified an average value of 20 W/(m2$
C) of comprehensive heat transfer coefficient and assumed that the heat transfer coefficient was a constant not changed with the time for simulating the temperature change curves at experimental measuring point (see Fig. 16(a)); Secondly, according to the principle of the isothermal section, the measured temperature curve was divided into n segments, and the dividing law projected onto the time axis was applied to the comprehensive heat transfer coefficient curve. The end points of each segment were defined as nodes, as shown in Fig. 16(b); Finally, based on the preset temperature field precision (d) and the determination order from the high temperature to the low temperature, the simulated air-cooling curves and the measured curves were fitted by adjusting the comprehensive heat transfer coefficient of the node (see Fig. 16(c)). In the process, the temperature difference between the simulated temperature Tsim and the experimental temperature Texp is needed, and if jTsimTexpj/Texpd %, which can be considered that the fitting of this section is good, otherwise, it needs to adjust the value of the system until all the node simulation values are close to the experimental values, then the simulation process is over (see Fig. 16 (d)). At this time, the comprehensive heat transfer coefficient under

Fig. 15. Flow chart for solving the comprehensive heat transfer coefficient under the condition of air cooling.

different air cooling parameters could be determined. 3.6.1. Material properties and simulation parameters Nowadays, the finite element method (FEM) simulation has been a frequently-used application tool that can dynamically visualize the data and adequately assist simulating materials forming processes to obtain the optimum technological conditions [16]. In this study, for software programs dealing with the aircooling process of magnesium alloy plate, the reliability of numerical simulation results primarily depends on the accuracy of the thermal-physical parameters input. It has been demonstrated that Java-based Materials Property (JMatPro) simulation software can be used to calculate the material data required in the finite element method (FEM) simulation for different types of materials (e.g. aluminium alloy, magnesium alloy, general steel) [17e20]. Thermal-physical parameters used in simulation have been calculated by JMatPro as functions of temperature in Fig. 17 taking all phases into account. FE simulation of the air-cooling process of magnesium alloy plate was realized using an implicit FE code DEFORM-3D with Lagrangian incremental simulation type, which was skilled in transient thermal analysis and has been widely used in the researches on Mg alloys [21e24]. In the finite element modeling, to simplify the analysis model, we ignored the role of the conveying roller in the process. In order to conveniently control the mesh density distributions and

Fig. 14. Comparison of cooling time between experimental and calculated values from different initial temperatures to 100
C (a, c) and 200
C (b, d).

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Fig. 16. Solving process of the comprehensive heat transfer coefficient.

element size in the plate, the brick-shaped element (hexahedron) was adopted for finite element mesh dividing. Considering that in the classic interface of DEFORM-3D, directly hexahedral meshes generating cannot be carried out, we used a commercial FEA preprocessing software HyperMesh as a grid processing tool. Then finite element models were imported into DEFORM-3D for the subsequent simulation. Because the surfaces (the upper surface, lower surface and side surfaces) of the plate are directly exposed to the external environment, the mesh density distributions in the plate were designed as a special form with gradually increasing element size from outside to inside so as to simplify and reduce the running time during the simulation, as shown in Fig. 18, which has been similarly reported by Sun [25] and Attarha [26]. 3.6.2. The governing equation and boundary conditions As mentioned in section 3.1 and 3.2, several factors may affect heat transfer during the air-cooling process of magnesium alloy

Fig. 17. Thermal-physical parameters versus test temperature calculated by JMatPro for the experimental AZ31B magnesium alloy.

plate, including heat conduction inside the plate and convection and radiation to the environment. We assumed that the thermal conductivity of the material was isotropic. Then, the energy balance equation of the heat transfer problem in the plate was:

      v vT v vT v vT vT KðTÞ$ þ KðTÞ$ þ KðTÞ$ ¼ rðTÞ$CðTÞ$ vx vx vy vy vz vz vt (16) which can be simplified as

KðTÞ$Tii  rðTÞ$CðTÞ$T_ ¼ 0

(17)

Where T is temperature of the plate, t is air-cooling time, K(T) is temperature-dependent heat transfer coefficient of plate, r(T) is temperature-dependent density, C(T) is temperature-dependent specific heat capacity. The temperature T of the plate is related to the time t in the unsteady state heat conduction problem, so the solution of Eq. (17) needs to have an initial condition, that is, the initial temperature

Fig. 18. Finite element meshes of AZ31B plate for air-cooling simulation.

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distribution at the beginning of the work piece. In this study, the initial temperature is a constant with T0, then

Tðx;

y;

z;

t ¼ 0Þ ¼ T0

(18)

The boundary condition for the heat transfer problem is as follows:

K ðT Þ


vT 4 ¼ Hcon ðTs  T∞ Þ þ srad ε Ts4  T∞ vn 
i h 2 ðTs þ T∞ Þ ðTs  T∞ Þ ¼ Hcon þ srad ε Ts2 þ T∞ ¼ hðTs  T∞ Þ

(19)

Where Hcon(TT∞) is thermal convection to the environment based on the Newton's law of cooling, sradε(T4T4∞) is thermal radiation to the environment based on the Stefan-Boltzmann's law, Ts is surface temperature, T∞ is ambient temperature, Hcon is convective coefficient to environment, srad is coefficient of Stefan-Boltzman, ε is emissivity of plate surface, and h is comprehensive heat-transfer coefficient.

3.6.3. Comparison of simulated and experimental values Under different initial temperatures and different initial thicknesses, the overall comprehensive heat transfer coefficients changed with the temperature of magnesium alloy plate during air cooling are as listed in Fig. 19. In order to verify the accuracy of the heat transfer coefficient, we compared the simulation results with the experimental results in detail (see Fig. 20). The results show that the maximum relative error between the simulated and the experimental values of the heat transfer coefficient is no more than 1%, indicating that heat transfer coefficients listed in Fig. 19 are very close to the actual values. In order to approximately estimate the value outside the range of experimental conditions, the least square method was used to fit the relationship between the heat transfer coefficient and experimental parameters. What is easy to see from Fig. 19 is that the dependence of comprehensive heat transfer coefficient of AZ31B magnesium alloy plate during air-cooling process on plate temperature and plate thickness is more significant than that on initial air-cooling temperature, especially, when it comes to the medium plate. In section 3.5, for the calculation of the temperature control model, we considered the heat loss was mainly caused by radiation and other factors were included in the change of emissivity, and based on the same assumption, J. Q. Sun [27] estimated the comprehensive heat transfer coefficient during the air-cooling process of steel plate as:

" h ¼ a1 $4:88$ε$

Ts þ 273 100

4

 #, Tex þ 273 4  ðTs  Tex Þ 100 

(20) where a1 is velocity correction factor. Combining Eq. (15) for determination of the emissivity ε and Eq. (20) for the estimate of the heat transfer coefficient h, we determined comprehensive heat transfer coefficient of AZ31B plate during the air-cooling transport process as h ¼ h (T, H). Meanwhile, the heat transfer coefficient has a very clear linear relationship on the plate temperature in the process, then the calculation equation of the heat transfer coefficient can be summarized as h(T, H)¼ k(H)$T þ b(H), where k(H) and b(H) is the slope and the intercept of the fitting line, respectively, and both of them have something to do with the thickness of the plate. Through non-linear fittings for the k(H) and b(H), we finally determined its relational formula as Eq. (21) by processing the data in Fig. 19.

Fig. 19. Heat transfer coefficients changed with the temperature under different thickness condition: (a) 2 mm, (b) 8 mm, (c) 15 mm, and (d) 31 mm.

Fig. 20. Comparison of the simulation air-cooling curves with the experimental curves under different thickness condition: (a) 2 mm, (b) 8 mm, (c) 15 mm, and (d) 31 mm.

hðT; HÞ ¼ 0:4361$H0:7 $T þ 6:947

(21)

Both the best-fitting of the heat transfer coefficient model (see Eq. (21)) and coefficient data determined by the above theory are shown in Fig. 21, from which a phenomenon can be observed that Eq. (21) can accurately fit the transfer coefficient data under different air-cooling conditions. And, Fig. 22 reveals that there is a good correlation between the calculated values and coefficient data with a 0.9882 R value. Although relatively large deviations exist, most of the relative errors are within the range of -10% to 10%, which indicates that the model established (see Eq. (21)) has a certain accuracy. 3.6.4. Further validation of heat transfer coefficient calculation model In order to reveal the regularity of the temperature distribution in the thickness direction, we have introduced an important nondimensional parameter of heat transfer, Biot number (Bi,

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calculated by Eq. (6)), by which quantitative study about the relationship between the thermal resistance of the heat conduction and the thermal resistance of the surface heat transfer can be carried out. It is easy to see from Fig. 17 that thermal conductivity (l) has a certain regularity along with the change of the plate temperature. Then, we used a third-order polynomial to characterize the influence of temperature on l (T) (see Eq. (22)), and its high fitting accuracy is shown in Fig. 23.

lðTÞ ¼ 194:2708  0:7926$T þ 0:0027$T 2  2:86  106 $T 3 (22) Bring Eqs. (22) and (21) into Eq. (6), we can get the Bi about different temperature (T) and thickness (H) (see Eq. (23)). According to the definition of the Biot number, MATLAB was used to draw the sketch-map of the Biot number for analyzing its distribution on the temperature and thickness, as shown in Fig. 24.

Bi ¼

Fig. 22. Correlation between data calculated by Eq. (21) and the data listed in Fig. 19.

0:4361$H0:7 $T þ 6:947 ðH=2Þ 194:2708  0:7926$T þ 0:0027$T 2  2:86  106 $T 3 (23)

From Fig. 24, what is easy to see is that with the increase of temperature and the increase of thickness, the Biot number has increased to some different extents: for temperature, it increased from (0.00008e0.0011) at 85
C to (0.0009e0.00275) at 400
C under different thicknesses; for thickness, it increased from (0.00008e0.0009) at 0.002 m to (0.00109e0.00275) at 0.031 m under different plate temperatures. This is a fact that a larger Biot number exists, indicating that the surface heat transfer is greater than the internal heat conduction, so there is a large difference in temperature between the surface and the inner of the plate along the conduction direction. Through comparative analysis of statistical Bi values, the temperature distribution law in the thickness direction under different initial temperatures and thicknesses (in section 3.3) can be explained. Because the value is obviously far less than 1, the temperature distribution law of the AZ31B plate should be in accordance with Type Ⅱ shown in Fig. 9(b), which explains the reason why the temperature distribution has an even higher degree of uniformity along the thickness. In general, as a result of that the temperature distribution law described by Bi is basically consistent with the experimental results, the accuracy of the Bi calculation formula is reflected, and the accuracy of the mathematical model of the comprehensive heat transfer coefficient is further proved.

3.6.5. Simulated verification of temperature distribution along the width direction Take 15 mm-350
C as an example, we comparatively analyzed the simulated and experimental values of the temperature at the position 4, 5, and 6 (in Fig. 3), meanwhile, we used mean absolute

Fig. 23. Fitting accuracy of thermal conductivity coefficient.

relative error (MARE) to evaluate the relationship between the simulated and experimental values, as show in Fig. 25. Observed from Fig. 25, the maximum MARE between the simulated and the experimental temperature values is no more than 3%, indicating that heat transfer coefficients input and simulated result are very close to the physical truth. Based on statistical simulation temperature data, we drew time-history curves of Tm and Tm', as shown in Fig. 26. From the figure, we can see that the extension tendency of the curves obtained from the simulation is almost the same as that of the experimental curves of 15 mm (250, 400
C) and 31 mm (250, 400
C) shown in Fig. 7, and there also exists an obvious peak point

Fig. 21. Comparison of heat transfer coefficient values listed in Fig. 19 and curved surface fitted by Eq. (21).

W. Jia et al. / Journal of Alloys and Compounds 695 (2017) 1838e1853

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Fig. 24. Distribution map of Bi value under different temperatures and thicknesses (a) and relationship of Bi between temperature (b) and thickness (c).

on each curve. Here we defined the stage before the peak point (Time: 0e135 s) as an inhomogeneity enhancement stage of temperature distribution and the stage after the peak point (Time: 161 s~) as a homogeneity enhancement stage. Through the previous analysis in section 3.2, the heat transfer mechanism is very different in the stage before and after the peak point, and then the tendency of temperature difference along the width direction during air-cooling process should be explained separately. After ignoring the heat conduction between the plate and conveying rollers, the symmetry condition along the directions of the width and length could be assumed, then we selected one section to observe and analyze. In order to study the temperature distribution along the width direction after a certain time period of air-cooling, we selected the time period of 10, 60, and 120 s to be the characterization of the earlystage (t < 135 s), and 200, 1000, and 1800 s to be the characterization of the late-stage (t > 161 s), as shown in Fig. 27. From Fig. 27, we can observe that the temperature increases gradually from the surface layer to the center layer along the thickness direction, and a same distribution trend exists from the edge side to the center of the width, which conforms to the temperature distribution law obtained by analyzing the experimental data in section 3.2 and 3.3. For the temperature distribution curves along the width direction in the figure, it is easily observed that from the center to the edge side, the curve first shows a relatively

Fig. 25. Comparison of the simulated air-cooling curves with the experimental curves at different positions along the width direction for 15 mm-350
C.

flat extension then a sharp drop in temperature. The transition point between the two trends is the so-called sudden-drop point. For the sudden-drop point in the paper, we made a detailed definition as follows. Firstly, in order to make a quantitative analysis of the temperature distributed along the width direction, we divided equally the 1/2 of width (1/2 W) into 16 segments, and the end points of each segment were defined as nodes (W ¼ 1, 2, 3, … 17, center and edge are 1 and 17, respectively). Secondly, we assumed that the temperature of plate had a linear decreasing trend from the center to the edge. Finally, first-order expressions for the dependence of both the test and the assumed temperature values on W at a certain moment were obtained, dT/dW, which was used to characterize the temperature changes between adjacent segments along the width direction. Take 15 mm-350
C-10 s (cooling-time period) as an example, dT/dW of the assumed temperature distribution curve is a constant, -0.1261 (negative value indicates a decrease in temperature distribution). However, for the simulation results, it is easy to find out that dT/dW extension with orientation changes decreases steadily first (dT/dW > 0.1261) and becomes sharply (dT/dW < -0.1261) later (see Fig. 28). Therefore, we could define that the point (dT/dW ¼ -0.1261) on the dT/dW extension curve was a sudden-drop point and then the orientation of the point along the width direction was obtained. The region in the range distanced from edge to the orientation of the sudden-drop point is the sudden-drop zone, about 0.1344 W for 15 mm-350
C after a cooling-time period of 10 s (see Fig. 28). We made a detailed statistic on the sudden-drop point after different cooling-time period (t  3 s) under different initial temperature and different thickness conditions (see Fig. 29). Observed from Fig. 29, as the air-cooling process continues, the sudden drop zone gradually expands from the edge to the central part. When the plate thickness is 15 mm, with the initial temperature increases from 300
C to 400
C, the temperature suddendrop region is extended from 0.2438 W to 0.2594 W. When the

Fig. 26. Time-history curves of Tm and Tm' for the simulation of 15 mme350
C.

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Fig. 27. Temperature distribution along the width direction after a certain time period of air-cooling: (a) 10 s, (b) 60 s, (c) 120 s, (d) 200 s, (e) 1000 s, (f) 1800 s.

plate initial cooling temperature is 350
C, with the plate thickness increases from 2 mm to 31 mm, the region is extended from 0.2313 W to 0.2626 W. To sum up, after a time period of 145e160 s, the range of the sudden-drop zone reaches its maximum, a range of 1/4 of width distanced from edge (orientation:9e17), which is a key research region for the study of the subsequent temperature compensation in the transport process. 4. Conclusion In this study, the as-cast AZ31B plate was subjected to the detection and tracking of temperature during the air-cooling transport process under a wide range of initial cooling temperatures (250e400
C) and plate thicknesses (0.002e0.031 m). Cooling law in the air-cooling process was analyzed through revealing the regularity of the temperature distribution in the thickness and the width directions and performing the quantitative analysis on the cooling parameters, such as cooling time, cool rate and heat transfer coefficient. The concrete results are as follows: (1) In the width direction, with the increase of the thickness or the initial cooling temperature, the maximum uniform degree of temperature distribution descends, and when the magnesium alloy plate is cooled by 25e56
C under different thickness and initial temperature conditions, inhomogeneous temperature distribution with the greatest degree will be generated. Then, in the AZ31B plate air-cooling process, the temperature drop of the plate overall should not exceed the range. And there is a sudden drop in temperature in the range of 1/4 of width distanced from edge, which is a key research area for the study of the temperature compensation. In the thickness direction, temperature distribution has a

Fig. 28. Search for the sudden-drop point and related orientation.

same law with that in the width direction about the thickness and initial temperature. But, the temperature difference of AZ31B plate in the thickness direction is much smaller than that in the width direction, which has been explained by the exact calculation of Bi number. (2) For the AZ31B plate with thin thickness and high initial temperature in the transport process of industrial production, improving the thermal insulation can achieve the precise control of temperature, however, for the thin plate with a lower initial temperature and for the medium plate, it can be effectively realized by considering the temperature drop calculation and increasing the preheating temperature before the air cooling, which also can availably reduce the loss and input required. (3) The emissivity of AZ31B plate has a very clear positive correlation on the initial air-cooling temperature and the thickness of the plate, especially presents a very clear linear relationship on the thickness. Therefore, the theoretical calculation formula of the emissivity was determined. After that, taking into account the heat transfer characteristics of magnesium alloy, the empirical formula of Stefan-Boltzmann was modified and optimized. Then, the temperature control model of AZ31B plate in air-cooling process was established with a high prediction accuracy, which is as follows:

Fig. 29. Related orientation on the sudden-drop point for 15 mm thickness under different initial temperatures (a) and for 350
C initial temperature under different thicknesses (b).

W. Jia et al. / Journal of Alloys and Compounds 695 (2017) 1838e1853

t¼

1853

i h
BHlrc 0:7364ðlnðT0 Þ  lnðT1 ÞÞ þ 0:002$ðT1  T0 Þ þ 0:8606  106 $ T02  T12 2εs  108 ðBl þ BH þ lHÞ

(4) The comprehensive heat transfer coefficient calculation method of air-cooling transport process under different conditions was performed by combining the finite element numerical simulation and the experimental results, then through fitting the relationship between the heat transfer coefficient and experimental parameters, theoretical calculation formula was defined accurately as:

hðT; HÞ ¼ 0:4361$H0:7 $T þ 6:947:

Acknowledgments This work is financially co-supported by the National Key Research and Development Program of China (2016YFB0301104) and the National Basic Research and Development Program of China (2013CB632203). References [1] W. Jia, S. Xu, Q. Le, L. Fu, L. Ma, Y. Tang, Modified FieldseBackofen model for constitutive behavior of as-cast AZ31B magnesium alloy during hot deformation, Mater. Des. 106 (2016) 120e132. [2] S. Mironov, T. Onuma, Y.S. Sato, H. Kokawa, Microstructure evolution during friction-stir welding of AZ31 magnesium alloy, Acta Mater. 100 (2015) 301e312. [3] J. Tu, S. Zhang, On the twinning growth mechanism in hexagonal close-packed metals, Mater. Des. 96 (2016) 143e149. [4] F. Feng, S. Huang, Z. Meng, J. Hu, Y. Lei, M. Zhou, Z. Yang, A constitutive and fracture model for AZ31B magnesium alloy in the tensile state, Mater. Sci. Eng. A 594 (2014) 334e343. [5] H.T. Jeong, T.K. Ha, Texture development in a warm rolled AZ31 magnesium alloy, J. Mater. Process. Tech. 187e188 (2007) 559e561. [6] W. Jia, L. Ma, Y. Jiang, P. Liu, H. Xu, Mathematical temperature field model about as-cast AZ31B magnesium alloy during hot rolling of plate, J. Rare Metal. Mat. Eng. 45 (3) (2016) 702e708. [7] W. Jia, L. Ma, Y. Tang, Q. Le, L. Fu, Relationship between microstructure and properties during multi-pass, variable routes and different initial temperatures hot flat rolling of AZ31B magnesium alloy, Mater. Des. 103 (2016) 171e182. [8] J. Zhang, A.T. Alpas, Transition between mild and severe wear in aluminium alloys, Acta Mater. 45 (1997) 513e528. [9] Y. Ding, Q. Zhu, Q. Le, Z. Zhang, L. Bao, J. Cui, Analysis of temperature distribution in the hot plate rolling of Mg alloy by experiment and finite element

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