Optimization of Roller Velocity for Quenching Machine Based on Heat Transfer Mathematical Model

During quenching process of steel plate, control parameters are important to product quality. In this work, heat transfer mathematical model has been developed for roller-type quenching machine to predict the temperature field of plate at first, and then an optimization schedule considering quenching technology and equipment limitations is developed firstly based on the heat transfer mathematical model with considering the shortest quenching time. A numerical simulation is performed during optimization process to investigate the effects of roller velocity on the temperature of representative plate. Based on the optimization method, study is also performed for different thickness of plate to obtain the corresponding roller velocity.The results show that the optimized roller velocity can be achieved for the roller-type continuous quenchingmachine based on the heat transfermathematical model. With the increasing of plate’s thickness, the optimized roller velocity decreases exponentially.


Introduction
Quenching is a common manufacturing process to obtain unique service performance properties of metal by controlling the amount and the morphology of microstructural constituents [1]. Roller-type continuous quenching machine is an important device which is commonly used in quenching heat treatment of plate in the iron and steel industry. In recent years, a number of works have been conducted on quenching machine referring to the heat transfer coefficient, critical cooling rate, plate temperature field, flow field, stress distribution, and microstructure distribution. Tang et al. [2], Şimşir and Gür [3], and Huiping et al. [4] simulated the temperature, stress, and microstructure field considering nonlinear material properties using finite element method during the quenching process. Kang and Im [5] established a 3D thermo-elastic-plastic model for plain-carbon steel along with phase transformation to improve the accuracy of numerical simulation. Zhu et al. [6] developed a threedimensional finite element flow field model of spraying pipe of roller-type quenching machine. Chantasiriwan [7] treated the problem of estimating the transient heat transfer coefficient at the surface of steel bars subjected to quenching using the sequential function specification method. Wang et al. [8] conducted a study on the heat transfer during quenching for plate roller quenching machine. Fu et al. [9] studied the critical cooling rate of plate during quenching. In their study, three types of quenching critical cooling rate models of medium carbon steel, low-carbon steel, and low-alloy construction steel were established with the modified Maynier model, Ei-dis model, and isothermal curve model, respectively.
In recent years, few researches on control parameters' optimization of continuous quenching machine were performed. Wang et al. [10] proposed a calculation method of optimal speed profile for online control system. Lü et al. [11] carried out a calculation model based on coupled metallothermo-mechanics theory. With the simulated results, they proposed an optimum quenching scheme. Huiping et al. [12,13] performed a step function model with respect to optimized technologic parameters during gas quenching process. The heat transfer plays an important role on the quenching heat treatment. However, studies on the optimization of control parameters based on the heat transfer model are few for the continuous quenching machine up to now. Therefore, in this work, the heat transfer mathematical model is developed during quenching process firstly. And then, an optimization method of roller velocity is proposed based on

Heat Transfer Mathematical Model
Heat transfer during quenching occurs via all possible heat transfer mechanisms, that is, conduction, convection, and radiation. Basically, heat is removed from the surface of the plate by convection to the water on the boundary during quenching process, which results in thermal gradients driving the conduction inside the plate. Because the surface temperature of the plate is higher than 800 ∘ C before quenching and higher than 100 ∘ C when the plate leaves the high-pressure water injection zone, the form of heat transfer between plate surface and cooling water is big tank boiling convection at the high-pressure water injection zone. The cooling water sprays on the surface of hot plate at constant velocity and angle. The result is that water near the hot plate's surface vaporizes quickly, taking the heat away in the form of latent heat of vaporization. The water away from the steel surface takes away the sensible heat through conduction and convection. When the plate passes the low-pressure water injection zone, the main heat transfer form is forced convection.

Governing Equations.
The transient heat transfer within the plate during quenching can be described mathematically by an appropriate form of Fourier's heat conduction equation (1). The equation can be expressed in its most general form as where ( ) is the thermal conductivity. ( ) is the density of material. ( ) is the constant pressure specific heat, and is quenching time.

Initial and Boundary Conditions.
The initial conditions at time = 0 can be described as where 0 is the initial temperature of plate. The boundary conditions are related to the heat transfer between quenching plate and cooling water. The relevant boundary conditions are given as where is the heat flux of plate's surface. ℎ Σ is the total heat transfer coefficient including convection and radiation. and are the thickness and width of the plate. in is the total quenching time.
is the water temperature. The total heat transfer coefficient ℎ Σ is related to the flow flux and velocity of the cooling water, the surface temperature, roughness and emissivity of the plate, and so on. The empirical equation is employed to express the total heat transfer coefficient which is available in [14,15], and it can be written as where , , and are regression coefficients. is the flow flux of the cooling water on the plate surface. is the surface temperature of the plate.

Numerical Method and Codes
Validation. The heat transfer mathematical model developed in this work can be used to predict the evolution of temperature during quenching process for the roller-type quenching machine. The governing equation (energy equation) with the boundary conditions is solved by finite volume method (FVM). The FVM is a common method to calculate the heat conduction problem. The codes validation on solving of energy equation has been done by the FVM in our former works which are available in [16,17].

Objective Function.
The cooling rate of plate during quenching process is determined by the roller velocity under constant water injection condition. Therefore, the roller velocity is the direct factor for quenching. The optimization of roller velocity plays an important role for the quenching machine. In this work, the optimization method of roller velocity is developed on the basis of the heat transfer mathematical model. The shortest quenching time is set as the optimization objective. Because the water injection condition is constant when the roller velocity is optimized, the shortest quenching time corresponds to the least water consumption. The objective function is given as where ℎ and are the length of high-pressure water injection zone and low-pressure water injection zone of the quenching machine. V is the roller velocity.

Constraints.
Constraints mainly include the quenching technology requirements and quenching machine equipment limitations. They can be described as  where and cr are the actual cooling rate of plate from 800 ∘ C to 500 ∘ C and critical cooling rate, respectively. V max and V min are the maximal velocity and the minimal velocity of roller. is the temperature at the centre of plate.
As the flow diagram shown in Figure 1, the roller velocity is optimized with the objective function in meeting the constraint conditions. Quenching under the optimized roller velocity corresponding to the shortest quenching time is also a process consuming the least water in the case of constant water injection condition.

Results and Discussion
A numerical analysis is performed firstly in this work to investigate the effects of roller velocity. Taking the typical plate 50 mm × 2750 mm × 12000 mm as simulated object, the central temperature of plate when it leaves high-pressure water injection zone and low-pressure water injection zone under the initial temperature 0 = 940 ∘ C is shown in Figure 2. It shows that the central temperature of plate at both positions decreases with decreasing of roller velocity. When the roller velocity decreases to 0.08 m/s, the central cooling rate of plate from 800 ∘ C to 500 ∘ C reaches 12 ∘ C/s, and the central temperatures of plate are 497 ∘ C and 29 ∘ C when it leaves high-pressure water injection zone and lowpressure water injection zone, respectively. These results meet the quenching technology requirements, so optimized rolled velocity (0.08 m/s) is obtained. During the optimization process, the central temperature of plate when it leaves highpressure water injection zone decreases slowly when the roller velocity is decreased from 1 m/s to 0.2 m/s and decreases quickly when the roller velocity is lower than 0.2 m/s. The decrease of plate central temperature when it leaves lowpressure water injection zone is faster than that when it   leaves high-pressure water injection zone as the roller velocity is decreased from 1 m/s to 0.2 m/s and is slower than that when it leaves high-pressure water injection zone as the roller velocity is decreased from 0.2 m/s. The temperature profiles of specified nodes of the typical plate are shown in Figure 3 under the optimized roller velocity (0.08 m/s) during quenching process. Because the heat flux is symmetric, 1/4 section of plate is selected to 167  show the temperature distribution as illustrated in Figure 4. The temperature of corner node decreases at the fastest rate, the central temperature of bottom surface is second, the central temperature of side surface is third, and the central temperature of the whole plate decreases at the lowest rate. There are three times temperature fluctuations of corner temperature and the central temperature of bottom surface because of the change of water flux. The temperature difference between bottom surface and central of the plate (line (1) and line (3) in Figure 3) increases at first and then decreases. The temperature difference is larger when the plate leaves high-pressure water injection zone and smaller when the plate leaves low-pressure water injection zone. The maximum temperature difference appears at the highpressure water injection zone, and the maximum value is about 660 ∘ C.
Taking this optimization method in this work, the roller velocity is also optimized for different thickness of plate. The thickness of plate is chosen as = 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, and 0.08 m to obtain optimized roller velocity and quenching time. The results are given as shown in Figure 5. As seen from the figure, with the increasing of plate's thickness, the roller velocity decreases exponentially. On the contrary, the quenching time increases dramatically. When the thickness of the plate increases from 0.01 m to 0.08 m, the corresponding roller velocity decreases exponentially from 0.97 m/s to 0.034 m/s, and the quenching time changes from 25.5 s to 729.4 s.

Conclusions
In this work, the temperature field of the plate is predicted by the heat transfer mathematical method. An optimization schedule is developed firstly with considering quenching technology and equipment limitations based on heat transfer Mathematical Problems in Engineering

Conflicts of Interest
The authors declare that they have no conflicts of interest.