- Original Article
- Open Access

# Controlling Roll Temperature by Fluid-Solid Coupled Heat Transfer

- Jing-Feng Zou
^{1, 2}, - Li-Feng Ma
^{1, 2}, - Guo-Hua Zhang
^{1, 2}Email author, - Zhi-Quan Huang
^{1, 2}, - Jin-Bao Lin
^{1}and - Peng-Tao Liu
^{1}

**31**:93

https://doi.org/10.1186/s10033-018-0293-7

© The Author(s) 2018

**Received:**5 June 2017**Accepted:**23 October 2018**Published:**7 November 2018

## Abstract

Currently, when magnesium alloy sheet is rolled, the method of controlling roll temperature is simple and inaccurate. Furthermore, roll temperature has a large influence on the quality of magnesium alloy sheet; therefore, a new model using circular fluid flow control roll temperature has been designed. A fluid heat transfer structure was designed, the heat transfer process model of the fluid heating roll was simplified, and the finite difference method was used to calculate the heat transfer process. Fluent software was used to simulate the fluid-solid coupling heat transfer, and both the trend and regularity of the temperature field in the heat transfer process were identified. The results show that the heating efficiency was much higher than traditional heating methods (when the fluid heat of the roll and temperature distribution of the roll surface was more uniform). Moreover, there was a bigger temperature difference between the input and the output, and after using reverse flow the temperature difference decreased. The axial and circumferential temperature distributions along the sheet were uniform. Both theoretical calculation results and numerical simulation results of the heat transfer between fluid and roll were compared. The error was 1.8%–12.3%, showing that the theoretical model can both forecast and regulate the temperature of the roll (for magnesium alloy sheets) in the rolling process.

## Keywords

- Magnesium alloy
- Fluid heating
- Heat transfer model
- Numerical simulation of fluid-solid coupling

## 1 Introduction

Magnesium alloy sheets have low density, good thermal conductivity, high specific strength and stiffness, desirable damping properties, and excellent electromagnetic shielding. Therefore they are widely used in the aviation, aerospace [1], automobile [2], high-speed rail, and electronics industries [3]. Consequently, magnesium alloy sheets have become the most promising non-ferrous metal material in the world today [4–6]. In the rolling process of magnesium alloy sheets, the roll temperature has a large influence on plate quality [7]. When the roll temperature is too low, the edges crack and surface cracking of the plate occurs. When the roll temperature is too high, the roll can become stuck [8, 9]. There is an urgent need to make an important breakthrough in the accurate control of roll temperature.

In recent years, the traditional methods for heating the roll have included the induction heating method, resistance wire heating method, and flame heating method [10, 11]. These traditional methods suffer from long heating times and uneven heating, therefore a new model using circular fluid flow to control roll temperature has been designed. This method heats and cools the roll using two tanks. The roll temperature can be controlled by adjusting the velocity and temperature of the fluid. A high precision heat transfer model (between the magnesium alloy sheet and roll) has been built, and the regulation strategy of roll temperature is given, meaning the roll temperature can be controlled within a reasonable range at any time. Therefore, this study can solve the existing problems of large crown, serious wave-shapes, and stuck rolls caused by inaccurate roll temperature control. This will significantly increase the finished production rate of the magnesium alloy sheets.

*D*= 320 mm,

*L*= 350 mm) is the chosen object to establish a fluid-solid coupled model of the fluid heating transfer roll. The roll temperature is studied for different times and velocities when fluid is heating the roll. The results are both an important and significant guide for controlling the roll temperature in the rolling process.

## 2 Design of the Roll Structure Using Fluid Heat Transfer

*M*) was less than 8% when the roll was designed. The roll bending rigidity before drilling is \(E\frac{{\uppi D^{4} }}{64},\) the smallest roll bending rigidity after drilling is \(E\frac{{\uppi D^{4} - 2n\uppi d_{0}^{4} - 16\uppi Qs^{2} d_{0}^{2} }}{64},\) the roll stiffness reduction is \(E\left( {\frac{{2n\uppi d_{0}^{4} }}{64} - Q\frac{{\uppi s^{2} d_{0}^{2} }}{4}} \right),\) and the roll bending strength reduced rate is \(M = \frac{{(2nd_{0}^{2} + 16Qs^{2} )d_{0}^{2} }}{{D^{4} }} \times 100\% \le 8\% .\)

Here, *E* is the modulus of elasticity of the roll (Pa); *d*_{0} is the diameter of thermostat oil hole (m); *D* is the diameter of the roll (m); *s* is the distance between the center of the thermostat hole and cross-sectional center of the roll (m); *n* is half the number of thermostat oil holes (the thermostat oil hole number is even). \(Q = \hbox{min} \left[ {\sum\limits_{k = 0}^{k = 2n - 1} {\left( {\sin \left( {x + \frac{\uppi}{n}k} \right)} \right)^{2} } } \right]\) (*k* is an integer, *x* is a real number and satisfies \(- \frac{\pi }{n} \le x \le \frac{\pi }{n},\) and the *Q* value can be obtained by mathematical software (MATLAB).

*d*

_{0}= 0.02 m, the distance between the center of the thermostat hole and the cross-sectional center of the roll was

*s*= 0.12 m, the number of thermostat oil holes was 8, and the roll bending strength reduced rate was 4.528%. This met the requirements. The two directions of the roll cross section in Figure 2 are shown in Figure 3(a) and (b). Fluid flows from oil hole 1, passed through oil holes 2‒7, then finally flowed from oil hole 8.

## 3 Establishment of the Analytical Model for Heat Transfer

The most accurate method for solving heat conduction is by mathematical analysis. However, the analysis can only solve the simple problem of heat conduction, whereas many of the complex thermal problems can only be obtained by numerical calculations using the finite difference method [16–19].

### 3.1 Hypothetical Condition

- (1)
The model of the fluid heat transfer roll is three-dimensional. The temperature is transmitted from the inner surface of the oil hole in the radial direction (the temperature difference in the circumferential direction can be neglected), so the model can be simplified as two-dimensional. This means the heat transfer is along the radial and axial directions, and the roll temperature field changes with time and position. Therefore, the model of the fluid heat transfer roll was simplified to a two-dimensional unsteady heat conduction model of a hollow cylinder.

- (2)
The physical property parameters of the roll and the fluid are not changed with time.

- (3)
The boundary conditions of fluid heat transfer, the roll inner and outer walls, and the outside are considered as the third boundary condition.

### 3.2 Calculation of Heating Transfer Process

*T*(

*r*,

*t*) changes with the time

*t*and position

*r*, the initial temperature distribution is

*T*(

*r*, 0) =

*T*(

*r*),

*r*

_{1}is the radius of the thermostat oil hole, and

*r*

_{2}is the closest distance from the thermostat oil hole center to the roll surface: \(r_{1} = \frac{{d_{0} }}{2} = 0.01\;{\text{m}},\)

*r*

_{2}= 0.04 m, and

*r*

_{1}<

*r*<

*r*

_{2}, as shown in Figure 3.

*f*

_{1}=

*h*

_{1}

*t*

_{f1};

*f*

_{2}=

*h*

_{2}

*t*

_{f2}.

*h*

_{1}is the heat transfer coefficient of the fluid and inner wall of the roll (W/(m

^{2}·°C));

*h*

_{2}is the heat transfer coefficient of the roll outer wall and the outside (W/(m

^{2}·°C));

*λ*is the thermal conductivity of the roll (W/m·°C));

*ρ*is the density of the roll (kg/m

^{3});

*c*is the specific heat capacity of the roll (kJ/kg);

*a*is the thermal diffusivity of the roll (m

^{2}/s);

*t*

_{f1}is the temperature of the fluid (°C); and

*t*

_{f2}is the outside temperature (°C).

*r*,

*t*) is divided into the grid of step Δ

*r*, Δ

*t*, as shown in Figure 4. Then there are:

*r*=

*r*

_{1}and

*r*=

*r*

_{2}, respectively for the forward difference and backward difference gives

*Δr*, and Eq. (1a) is differentially calculated to give

*Δr*)

^{2}+ (

*Δt*), the right side of Eq. (10) includes \(T_{0}^{n}\) when

*j*= 1 and \(T_{N}^{n}\) when

*j*

*=*

*N*− 1. These temperatures can be calculated by Eqs. (7) and (8), the superscript

*n*

*+*1 in place of

*n*, and then the resulting \(T_{0}^{n}\) and \(T_{N}^{n}\) put into Eq. (10). The results are summarized as follows:

*n*= 0, 1, 2,… and \(\beta_{i} = 1 + \frac{{h_{i} \Delta r}}{\lambda },\) \(\gamma_{i} = \frac{{f_{i} \Delta r}}{\lambda },\) and

*i*= 1 or 2.

Equations (11) to (14) give the finite difference expression of the heat conduction equation. The solving method for this is described below.

Equation (12) provides *N* − 1 algebraic equations to solve the *N* − 1 unknown nodal temperatures: \(T_{1}^{n + 1} ,\;T_{2}^{n + 1} , \ldots ,T_{N - 1}^{n + 1} .\) Due to the use of explicit form these equations are not coupled, and may be calculated separately. Calculation is started with *n* = 0, \(T_{1}^{1} ,\;T_{2}^{1} , \ldots ,T_{N - 1}^{1}\) and are determined according to Eq. (12), then the temperature values are found when *n* = 1,2,…. As \(T_{1}^{n + 1}\) and \(T_{N - 1}^{n + 1}\) can be obtained by Eq. (12), then the boundary surface temperature \(T_{0}^{n + 1}\) and \(T_{N}^{n + 1}\) can be calculated respectively according to Eqs. (11) and (13).

*a*and Δ

*r*are determined, the time step Δ

*t*will be limited by the stability criterion:

### 3.3 Calculation of the Heat Transfer Coefficient

When the size of the hole and type of fluid are determined, the heat transfer coefficient between the fluid and the inner wall is related to the temperature and velocity of the fluid. In this paper, the heat conducting fluid was L-QD330, and the initial temperature of the fluid was 300 °C. In the following section the effect of fluid velocity on heat transfer coefficient will be studied. The faster the fluid the greater the heat transfer coefficient, but the flow resistance will increase when the velocity increases, and damage to the equipment will increase. To protect the equipment and reduce pipe wear, the velocity was controlled within a reasonable range.

Physical property parameters of heat conducting fluid L-QD330

Temperature (°C) | Density (kg/m | Specific heat (kJ/kg) | Heat conductivity coefficient (W/m·°C) | Kinematic viscosity (mm |
---|---|---|---|---|

20 | 864.8 | 1.669 | 0.139 | 72.0 |

200 | 775.0 | 2.490 | 0.124 | 1.15 |

220 | 765.0 | 2.512 | 0.121 | 0.80 |

250 | 754.9 | 2.658 | 0.119 | 0.52 |

270 | 740.0 | 2.706 | 0.117 | 0.46 |

300 | 724.9 | 2.823 | 0.116 | 0.37 |

*Pr*number is \(Pr_{{_{f} }} = \frac{{c\rho v_{f} }}{\lambda }\) [20, 21], the

*Nu*number is \(Nu_{f} = 0.027Re_{f}^{0.8} Pr_{f}^{1/3} \left( {\frac{{u_{f} }}{{u_{w} }}} \right)^{0.25} ,\) and when

*Re*> 10

^{4}the fluid flow regime is turbulent [22, 23]. Therefore, the heat transfer coefficient is \(h_{1} = \frac{{\lambda Nu_{f} }}{d},\) where

*u*

_{m}is the velocity of the fluid (m/s);

*v*

_{f}is the kinematic viscosity of the fluid (m

^{2}/s);

*c*is the specific heat of the fluid (kJ/kg);

*ρ*is the density of the fluid (kg/m

^{3});

*λ*is the thermal conductivity of the fluid (W/m·°C);

*u*

_{f}and

*u*

_{w}are the fluid dynamic viscosities under fluid temperature

*t*

_{f}and wall temperature

*t*

_{w}as characterization temperature (Pa·s); \(t^{\prime\prime}_{f}\) is the fluid export temperature when \(t^{\prime}_{f}\) is the qualitative temperature, and \(\overline{M} = \rho \frac{{\uppi d^{2} }}{4}u_{m}\) is the traffic quality of the fluid.

Heat transfer coefficient and the average fluid temperature of the respective velocities after calculation

Velocity (m/s) | 0.3 | 0.5 | 0.7 | 0.9 | 1.1 |

| 597.4 | 1096.1 | 1489.4 | 1770.5 | 2055.9 |

| 260.6 | 268.3 | 275.5 | 284.5 | 290.5 |

*λ*,

*v*, and

*Pr*are the heat conductivity coefficient (W/m °C), the kinematic viscosity (W/m °C), and the

*Pr*number of the outside when the temperature is

*t*

_{m}, respectively. Further,

*T*is the average temperature of the roll (°C);

*σ*is 5.669 × 10

^{−8},

*ε*is the emissivity of the roll, and

*t*

_{f2}is the outside temperature (20 °C).

*h*

_{1}) is calculated in the ideal circumstances of a constant wall temperature. In fact, the wall temperature gradually increased, the temperature and physical parameters of the fluid were always changing, and the heat transfer coefficient reduced. Therefore we needed to correct the calculated values. Table 3 describes the heat transfer coefficient of the respective velocities after the experiment. Calculation and experimental results are shown in Figure 5. The heat transfer coefficient was corrected, and the fitting equation is shown as Eq. (20). The heat transfer coefficient has more influence than the fluid temperature on the heat transfer process, so the temperature of the fluid is the average calculating value.

Heat transfer coefficient of the respective velocities after the experiment

Velocity (m/s) | 0.3 | 0.5 | 0.7 | 0.9 | 1.1 | |
---|---|---|---|---|---|---|

Heat transfer coefficient (W/(m | 1 min | 2005.05 | 2490.66 | 2560.31 | 2670.53 | 2782.13 |

2 min | 1468.65 | 1516.04 | 1441.64 | 1432.77 | 1426.73 | |

3 min | 993.25 | 901.58 | 786.50 | 768.58 | 595.94 | |

4 min | 692.73 | 572.07 | 508.20 | 491.74 | 396.19 | |

5 min | 496.52 | 433.25 | 365.03 | 373.41 | 280.41 | |

6 min | 440.29 | 363.71 | 313.75 | 312.08 | 238.21 | |

7 min | 390.37 | 321.21 | 279.03 | 298.76 | 211.17 | |

8 min | 365.11 | 294.54 | 261.38 | 248.12 | 195.70 | |

9 min | 341.47 | 271.79 | 248.76 | 224.61 | 188.87 | |

10 min | 329.66 | 254.77 | 239.66 | 210.50 | 181.30 |

*h*

_{1}and

*h*

_{2}were generated into the equation at various velocities, and then the roll temperature was calculated under various velocities over time.

## 4 Discussion of Numerical Simulation and Results

### 4.1 Numerical Simulation

^{−3}mm, and the minimum value was 5

^{−4}mm. Due to the viscosity of the heat conduction oil during the flow process, five layers of fluid boundary layer were set up, and the growth rate was 1.1, as shown in Figure 6(b). The heat transfer model was set as transient, the temperature unit was K, the fluid flow state was set to turbulent state, and the roll parameters were set as shown in Table 4. The initial temperature of the roll was 20 °C, and the heat transfer process of the velocity at 0.3, 0.5, 0.7, 0.9 and 1.1 m/s were numerically simulated.

Physical property parameters of the fluid and roll

Material | Density (kg/m | Specific heat (kJ/kg) | Heat conductivity coefficient (W/m·°C) | Kinematic viscosity (mm |
---|---|---|---|---|

Fluid | 735 | 2.75 | 0.117 | 0.37 |

Roll | 7800 | 0.86 | 49.8 | – |

### 4.2 Results Analysis of Numerical Simulation

*t*) and the pressure difference (Δ

*P*) between the import and export (when the roll was heated from room temperature to 200 °C under different velocities) are shown in Table 5. The trend of the change is shown in Figure 10. The fluid velocity changed from 0.3 to 1.1 m/s, t reduced from 34.4 to 15.5 minutes, the pressure increased from 0.03 to 0.32 MPa, the heating time shortened, and the velocity increased to a certain value. Further, the heat transfer increase rate decreased, but the pressure became larger and the flow resistance increased. The relationships between velocity (

*v*), time (

*t*), and the pressure difference (Δ

*P*) between the import and exports were fitted as in Eqs. (21) and (22). In the actual production, the best velocity was selected according to the required heating time, and whether different equipment could withstand the pressure.

Time and pressure values up to 200 °C under different velocities

Velocity (m/s) | 0.3 | 0.5 | 0.7 | 0.9 | 1.1 |

Time (min) | 34.4 | 24.5 | 21.3 | 16.2 | 15.5 |

Pressure (MPa) | 0.03 | 0.07 | 0.12 | 0.20 | 0.32 |

### 4.3 Results Analysis of Theoretical Calculations

### 4.4 Comparative Analysis of Theoretical and Numerical Data

## 5 Heat Transfer after Reverse Flow of Fluid

Equilibrium time and equilibration temperature after different flowing times

Flow time (min) | 3 | 5 | 7 | 9 | 11 |

Equilibrium time (min) | 4 | 4 | 4 | 4 | 5 |

Equilibrium temperature (°C) | 153 | 172 | 187 | 202 | 216 |

The results showed that the equilibrium time was substantially constant when the positive fluid flow was longer, but the equilibrium temperature increased with the velocity increase. After the reverse flow of fluid, the export temperature of the roll increased, but the length of this area range was less than 1/5 of the length of the roll, and the temperature difference (between the surface area of the hole and the surface area of non-holes) did not exceed 8 °C. The axial temperature difference at the roll import was approximately 18 °C, the high temperature zone was concentrated at 1/8 of the roll body length, and the temperature of other positions was uniform. The surface temperature for the remaining 7/8 of the roll surface was even more uniform. As the roll surface temperature difference was small, the rolling process could be carried out.

*t*) and equilibrium temperature (

*T*) is fitted we get:

Therefore we can determine the fluid flowing time according to the required roll temperature to obtain a more accurate roll temperature value.

## 6 Conclusions

- (1)
The theoretical calculation results and numerical simulation results of heat transfer between the fluid and roll were compared (the error was 1.8%‒12.3%). This showed that the heat transfer theoretical model could forecast and regulate roll temperature when magnesium alloy sheets are rolled.

- (2)
Compared with the traditional heating method, the method of heating the surface temperature of the roll with the fluid reduced the heating time, and the temperature distribution of the roll surface was more uniform. However, the temperature difference between import and export was bigger. The lager the velocity, the faster the heat transfer, but the rate of increase decreased.

- (3)
The method of fluid reverse flowing after a period of positive flowing was used to balance the temperature difference between import and export. The longer the time of fluid flow, the time required to reach equilibrium was constant (approximately four minutes). When the fluid forward flow time was longer, the equilibrium temperature was higher. The more accurate roll temperature can be obtained by reasonably determining the time of forward flow of fluid.

## Declarations

### Authors’ Contributions

J-FZ completed the numerical simulation, data analysis and the modification of the manuscript; L-FM guided the whole data analysis process and approved the final version; G-HZ derived the formulas; Z-QH, J-BL carried out literature search and data acquisition; P-TL reviewed and edited the manuscript. All authors read and approved the final manuscript.

### Authors’ Information

Jing-Feng Zou, born in 1993, is a graduate candidate at *Taiyuan University of Science and Technology*, *China.* Her research interests include roller temperature field control.

Li-Feng Ma, born in 1977, is currently a professor at *Taiyuan University of Science and Technology*, *China.* He received his PhD degree from *Taiyuan University of Technology*, *China*, in 2008. His research interests include magnesium alloy sheet properties study.

Guo-Hua Zhang, born in 1991, is a graduate candidate at *Taiyuan University of Science and Technology*, *China.* Her research interests include roller temperature field control.

Zhi-Quan Huang, born in 1982, is currently an associate professor at *Taiyuan University of Science and Technology, China.* He received his PhD degree from *Taiyuan University of Science and Technology*, *China*, in 2014. His research interests include rolling process and equipment design, light alloy deformation process theory and technology.

Jin-Bao Lin, born in 1979, is currently a professor at *Taiyuan University of Science and Technology*, *China.* He received his PhD degree from *Shanghai Jiao Tong University*, *China*, in 1996. His research interests include magnesium alloy plastic forming, large plastic deformation technology.

Peng-Tao Liu, born in 1989, is currently a PhD candidate at *Taiyuan University of Technology*, *China.* His research interests include Composite plate shape and performance.

### Acknowledgements

The authors sincerely thanks to Professor Da-Qing Fang of Xi’an Jiao Tong University for guidance on the usage of old equipment in the field, and to Professor Qing-Xue Huang of Taiyuan University of Technology for valuable discussion.

### Competing Interests

The authors declare that they no competing interests.

### Funding

Supported by National Natural Science Foundation of China (Grant No. U1510131), Key Research and Development Projects of Shanxi Province of China (Grant Nos. 201603D121010, 201603D111004), Science and Technology Project of Jin Cheng City of China (Grant No. 20155010), Youth Program of National Natural Science Fund of China (Grant No. 51604181), Project of Young Scholar of Shanxi Province, Leading Talent Project of Innovative Entrepreneurial Team of Jiangsu Province (Grant No. 51501122).

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## Authors’ Affiliations

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