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Heat Mass Transfer

DOI 10.1007/s00231-015-1602-7

ORIGINAL

Solution of boundary heat transfer coefficients between hot stamping die and cooling water based on FEM and optimization method

Huiping Li1 · Lianfang He1 · Chunzhi Zhang1 · Hongzhi Cui1

Received: 8 September 2014 / Accepted: 30 May 2015 © Springer-Verlag Berlin Heidelberg 2015 x, y and z Cartesian coordinates n Outer normal of boundary surface

Hk Convection coefficient

Hs Radiation coefficient

Tw Temperature of boundary

Tc Temperature of ambience

H Boundary heat transfer coefficient (BHTC)

E(x) Function of standard deviation f Absolute value of E(x) m Total number of target points

Ti Temperature of ith target point measured by sensor

Tˆi Temperature of ith target point evaluated by numerical simulation α Step size of searching the interval

E(a) Value relative to the left end-point of interval [a, b]

E(b) Value relative to the right end-point of interval [a, b] 1 Introduction

In order to reduce the spring-back and forming force of ultra high strength steels, hot stamping was presented in recent years, and some researchers have been studying the numerical simulation of hot stamping. In the simulation of hot stamping, the thermal physical parameters have significant effects on the calculation accuracy of physical fields, and the boundary heat transfer coefficient (BHTC) in the hot stamping is one of the most important thermal physical parameters. The solution of BHTC is one of the inverse heat conduction problems (IHCP). It is an ill-posed problem and more difficult to solve than the normal heat exchange problem. For IHCP, some sensors laid at some

Abstract The thermal physical parameters have significant effects on the calculation accuracy of physical fields, and the boundary heat transfer coefficient between the die and water is one of the most important thermal physical parameters in the hot stamping. In order to attain the boundary heat transfer coefficient, the testing devices and test procedures are designed according to the characteristic of heat transfer in the hot stamping die. A method of estimating the temperature-dependent boundary heat transfer coefficient is presented, and an inverse heat conduction software is developed based on finite element method, advance-retreat method and golden section method. The software is used to calculate the boundary heat transfer coefficient according to the temperatures measured by NiCr–NiSi thermocouples in the experiment. The research results show that, the convergence of the method given in the paper is well, the surface temperature of sample has a significant effect on the boundary heat transfer coefficient between the die and water. The boundary heat transfer coefficient increases as the surface temperature of sample reduces, and the variation is nonlinear.

List of symbols k Thermal conductivity

T Temperature qv Latent heat of phase-transformation ρ Density cp Constant pressure specific heat t Time * Huiping Li lihuiping99@163.com 1 School of Materials Science and Engineering, Shandong

University of Science and Technology, 579 Qianwangang

Road, Qingdao 266590, Shandong, People’s Republic of China

Heat Mass Transfer 1 3 certain positions should be used to record the temperatures of part. The unknown conditions, such as BHTC and boundary temperature, can be reckoned by some special methods according to the temperature variation.

Many researchers have studied the solution of IHCP, and presented some useful methods. According to the temperature–time data of several interior locations in the quenching part measured by sensors, Gu et al. [1] used the inverse heat conduction method to estimate the heat transfer coefficients between quenching part and water or oil. Based on the Karhunen–Loeve Galerkin procedure, Park et al. [2] presented a method for the solution of inverse heat conduction problems of estimating the time-varying strength of a heat source in a two-dimensional heat conduction system.

Taler and Zima [3] used the control volume method to solve multi-dimensional inverse heat conduction problems.

Chantasiriwan [4] studied the one-dimensional problem of estimating the transient heat transfer coefficient at the surface of steel bars using the sequential function specification method. Chen et al. [5] estimated the heat transfer coefficients between quenching part and quenching medium using the inverse heat conduction method. Li et al. [6] calculated the temperature-dependent BHTC in the quenching, and the phase-transformation latent heat is considered to improve the calculation precision of temperature.

Moreover, Lesnic, Kim, Taler, Shen and Tseng respectively presented the boundary element method [7], decomposition method [8], integral method [9], Tikhonov’s regularization method [10], direct sensitivity coefficient method [11] to solve the inverse heat conduction problems.

In the hot stamping, there are two types of BHTC, one is the BHTC between the boron steel plate and hot stamping tools, and the other is the BHTC between the flowing water and hot stamping tools. Hot stamping can be used to produce the components with distributed microstructure and mechanical properties by using the slow cooling rate for the special region in hot stamping [12]. Local reductions of cooling rate can be achieved by using the thin air gap between the tools and the blank [13], controlling the temperature of tools [14], or using the steel with the lower thermal conductivity in the tools [15]. The production of structural parts with distributed mechanical properties requires a precise control of the local cooling rate and temperature of tools [16]. This can only be achieved with an accurate knowledge of the two type of BHTC in the hot stamping.

Some researchers have studied the BHTC between the boron steel plate and tools. Bosetti et al. [17] calculated the

BHTC between the steel sheets and tools under conditions very close to the industrial ones. Abdulhay et al. [18, 19] designed an experimental device to accurately measure the thermal contact resistance under representative process conditions, and presented an approach to determine the evolution of the thermal contact resistance under different contact pressure (2–30 MPa). The research results of Li et al. [20] show that, the oxidation of boron steel has a remarkable effect on the surface heat transfer coefficient, the surface heat transfer coefficient increases with the rise of boundary pressure, and the relationship is approximately linear. All above research results show the BHTC between the boron steel plate and tools depends on the pressure. Some other research results show the BHTC between the boron steel plate and tools also depends on boundary temperature difference [21].