熔模铸造条件下Ti6Al4V合金铸件与陶瓷型壳间界面换热系数研究<sup>*</sup>

引用本文

邵珩, 李岩, 南海, 许庆彦. 熔模铸造条件下Ti6Al4V合金铸件与陶瓷型壳间界面换热系数研究^* . 2015, 51(8): 976-984[Heng SHAO, Yan LI, Hai NAN, et al. RESEARCH ON THE INTERFACIAL HEAT TRANSFER COEFFECIENT BETWEEN CASTING AND CERAMIC SHELL IN INVESTMENT CASTING PROCESS OF Ti6Al4V ALLOY[J]. AMS, 2015, 51(8): 976-984] 复制到剪切板

Permissions

熔模铸造条件下Ti6Al4V合金铸件与陶瓷型壳间界面换热系数研究^*

邵珩¹, 李岩², 南海², 许庆彦¹

1 清华大学材料学院先进成形制造教育部重点实验室, 北京 100084

2 北京航空材料研究院, 北京 100095

邵珩, 1989年生, 男, 博士生

收稿日期:2015-05-02

基金:* 中欧航空科技合作项目, 欧盟第七框架项目, 国家重点基础研究发展计划项目 2011CB706801, 国家自然科学基金项目51171089和51374137, 国家重大科技计划项目2012ZX04012011, 以及国家高技术研究发展计划项目2007AA04Z141资助;

摘要

建立了Ti6Al4V合金铸件/铸型界面换热系数(h)的一维反算模型, 从数学及数值模拟的角度研究了型壳热物性参数和热电偶定位等参数对h计算的影响, 分析了不同参数影响的不同特点, 据此对型壳热物性参数和热电偶定位位置等进行了修正, 提高了h反算精度. 修正计算参数后的反算结果表明, Ti6Al4V合金熔模铸造条件下, h的变化可分为4个阶段: (1) 铸件为液态, h维持约440 W/(m²K); (2) 铸件表面生成完整凝固层, 此阶段h下降近60%; (3) 凝固层不断增厚至铸件凝固, 此阶段h下降接近峰值的20%; (4) 铸件凝固后, h随温度缓慢下降. 在三维模型中对反算得到的h进行了验证, 得到的模拟温度与实测温度基本吻合, 表明反算得到的h较为准确, 可以应用于Ti6Al4V合金熔模铸造过程的数值模拟中.

关键词: 钛合金; 熔模铸造; 界面换热系数

中图分类号:TG249.5

RESEARCH ON THE INTERFACIAL HEAT TRANSFER COEFFECIENT BETWEEN CASTING AND CERAMIC SHELL IN INVESTMENT CASTING PROCESS OF Ti6Al4V ALLOY

Heng SHAO¹, Yan LI², Hai NAN², Qingyan XU¹

1 Key Laboratory for Advanced Materials Processing Technology, Ministry of Education, School of Materials Science and Engineering, Tsinghua University, Beijing 100084

2 Beijing Institute of Aeronautical Materials, Beijing 100095

Correspondent: XU Qingyan, professor, Tel: (010)62795482, Email:scjxqy@mail.tsinghua.edu.cn

Fund:Supported by China-EU (European Union) Science & Technology Cooperation in Aviation, 7th Framework Programme of EU, National Basic Research Program of China (No.2011CB706801), National Natural Science Foundation of China (Nos.51171089 and 51374137), National Science and Technology Major Project (No.2012ZX04012011) and High Technology Research and Development Program of China (No.2007AA04Z141);

Abstract

Investment casting process is an important way to get complex parts of titanium alloy. However there are few research on the interfacial heat transfer coefficient (h) between casting and shell thus the temperature simulation of investment casting process of titanium alloy is often inaccurate. In order to get a relatively accurate h, a one-dimensional mathematical model for the reverse calculation of h between casting and shell in investment casting process of Ti6Al4V alloy was built and the analytic relationship between temperature and time/heat flux was established. Considering the calculated h is significantly affected by the error of parameters such as the specific heat capacity and thermal diffusivity of shell and position of thermocouples, research on the error of these parameters is essential. The relationship between the error of these parameters and the temperatures in the casting and shell was studied and it was found that the effect of different kind of error on the temperature field was obviously different. An experiment based on the one-dimensional mathematical model was done and temperatures of different positions were measured. Based on the effect of different kind of error and the difference between the calculate temperature field and the measured temperatures, the proportion of effect of each kind of error was assessed. These errors were revised on the basis of the assessment, thus a relatively accurate h between the casting and shell was obtained. The relationships between h and thickness of the solidified layer on the casting/temperature at the surface of casting can be divided into 4 stages: (1) Metal was liquid and h kept about 440 W/(m²K); (2) Solid layer appeared on the surface, and h declined nearly 60%; (3) Solid layer grew up before metal became completely solid and h declined nearly 20% of its maximum; (4) After metal solidified, h declined slowly as temperature on the surface of casting dropped. These relationships were applied in a three-dimensional model for numerical simulation of the temperature field. Temperatures of different positions in casting and shell were calculated and calculated temperatures agreed with measured ones well. Thus the accuracy of h was identified and it can help solve problems in the production in investment casting process of Ti6Al4V alloy.

Keyword: titanium alloy; investment casting; interfacial heat transfer coefficient

Show Figures

钛合金具有比强度高、耐热性好、耐蚀性好、生物相容性好等优点, 因而被广泛应用于航空航天、医疗及化学工业. 但钛合金零件往往具有十分复杂的形状, 难于进行机械加工, 因而需要采用熔模铸造工艺来成形. 在熔模铸造工艺开发过程中, 数值模拟因为能够预测缺陷的形成, 减少工艺开发所需的时间和成本而受到青睐, 得到越来越多的应用^[1,2,3]. 但数值模拟的精确度容易受到相关模拟参数精确度的直接影响, 如铸件和铸型的热物性参数、铸件铸型间的界面换热系数等. 通常而言, 前者可以通过查阅文献或实验测量得到较为准确的数值, 后者都是通过估计给出, 且经常作为常数使用^[1,2], 可靠性和精确性都有待商榷. 因此, 对熔模铸造条件下钛合金铸件与铸型界面换热系数的专门研究就显得尤为必要.

近年来, 国内外学者对铸造过程铸件/铸型界面换热系数的研究, 主要涉及压铸^[4,5,6,7,8]、熔模铸造^{[9,10,11,12,13,14,15]}、挤压铸造^[16,17,18]、砂型铸造^[19,20]和连续铸造^[21,22]等, 包含铝合金、镁合金、高温合金和钢等, 取得了较好的效果. 郭志鹏等^[4,5,6,7]针对铝合金压铸条件下的界面换热行为展开了大量的研究, 建立了完善的数值模型, 分析了铸件厚度、压铸压力、铸型温度和测量位置等对界面换热系数的影响. Konrad等^[11]针对IN738LC高温合金熔模铸造条件下的界面换热行为, 测量了陶瓷型壳的比热容和热扩散率, 研究了型壳预热温度和厚度对铸件晶粒尺寸和二次臂间距的影响, 并反算得到各工艺参数(型壳预热温度和型壳厚度)下铸件凝固过程中界面换热系数随铸件温度的变化. Sun等^[16]研究了挤压铸造条件下AM60 镁合金与金属铸型之间的界面换热行为, 比较多项式拟合方法与反算法计算得到的界面换热系数, 结果表明反算法的精度较高. 徐宏等^[19]研究了砂型铸造条件下ZL104与铸型间的界面换热系数, 给出了界面换热系数随时间的变化. Wang等^[22]测量了连续铸造条件下到弯月面不同距离处的钢材表面温度, 反算得到连续铸造过程中的界面换热系数, 并以此预测不同位置钢材表面凝壳厚度, 预测结果与实测结果吻合良好.

钛合金熔点高, 化学活性高, 多采用熔模铸造工艺, 铸型采用陶瓷材料, 铸造过程在真空炉中完成^[23,24], 大大增加了实验设计与数据处理的难度, 目前尚鲜有对其铸造过程铸件/铸型界面换热系数的研究, 无法为生产过程提供更多的帮助. 因此, 本工作拟采用热传导反算法, 测算Ti6Al4V合金熔模铸造铸件/型壳界面换热系数, 以期为Ti6Al4V合金熔模铸造过程数值模拟提供准确的计算参数.

1 物理/数学模型

1.1 铸件/铸型界面换热物理模型

本工作采用平板形铸件, 测量平板中央部位型壳的温度. 考虑到铸件高度与宽度远大于厚度, 因此将平板中央部位型壳的温度场近似视为随距铸件/型壳界面的距离而变化的一维温度场, 如图1所示.

	Figure Option View Download New Window
	Fig.1 Schematics of one-dimensional heat transfer model (a) and temperature field in the model (b) (q₁—interfacial heat flux between casting and mold; q₂—heat flux across the surface of mold; x—distance from the interface between casting and mold; d—thickness of mold; T—temperature)图1 一维传热模型及其温度场示意图

在型壳中, 其温度T分布为:

(1)	$\begin{matrix} T = T (x, τ, q_{1}, q_{2}) \end{matrix}$

(2)	$\begin{matrix} q_{1} = q_{1} (τ) \end{matrix}$

(3)	$\begin{matrix} q_{2} = q_{2} (τ) \end{matrix}$

图1及式(1)~(3)中, T为型壳中不同位置、不同时刻的温度, x为到铸件/型壳界面的距离, τ为时间, q₁为铸件/型壳界面的热流密度, q₂为型壳外表面热流密度.

1.2 铸型温度分布数学模型

在简化后的一维传热模型中, 铸型内的温度分布满足以下条件:

(4)	$\begin{matrix} ρ c \frac{? T}{? τ} = \frac{?}{? x} (λ \frac{? T}{? x}) \end{matrix}$

(5)	$\begin{matrix} T (x, 0, q_{1}, q_{2}) = T_{0} \end{matrix}$

(6)	$\begin{matrix} λ \frac{? T}{? x} \|_{x = 0} = - q_{1} (τ) \end{matrix}$

(7)	$\begin{matrix} λ \frac{? T}{? x} \|_{x = d} = - q_{2} (τ) \end{matrix}$

式(4)~(7)中, $\begin{matrix} ρ, c \end{matrix}$ 和 $\begin{matrix} λ \end{matrix}$ 分别为型壳材料的密度、比热容和导热系数; $\begin{matrix} T_{0} \end{matrix}$ 为型壳初始温度.

对陶瓷型壳热物性参数的实际测量结果表明, 型壳的 $\begin{matrix} c \end{matrix}$ 与 $\begin{matrix} λ \end{matrix}$ 随温度的变化均较小. 忽略温度对型壳热物性参数的影响, 将 $\begin{matrix} ρ, c 和 λ \end{matrix}$ 作为常数处理, 式(4)可简化为:

(8)	$\begin{matrix} ρ c \frac{? T}{? τ} = λ \frac{?^{2} T}{? x^{2}} \end{matrix}$

暂令 $\begin{matrix} q_{2} \end{matrix}$ =0, 仅考虑 $\begin{matrix} q_{1} \end{matrix}$ 对T的影响, 则有:

(9)	$\begin{matrix} T = T (x, τ, q_{1}) \end{matrix}$

(10)	$\begin{matrix} T (x, 0, q_{1}) = T_{0} \end{matrix}$

(11)	$\begin{matrix} λ \frac{? T}{? x} \|_{x = d} = 0 \end{matrix}$

式(9)中的T满足式(6), (8), (10)和(11)的各项条件.

假设对某一特定的热流 $\begin{matrix} q_{1}^{*} \end{matrix}$ , 当 $\begin{matrix} q_{1} = q_{1}^{*} \end{matrix}$ 时, $\begin{matrix} T = T^{*} \end{matrix}$ 为满足式(6), (8), (10)和(11)的解. 将 $\begin{matrix} q_{1}^{*} \end{matrix}$ 任意分拆为2股热流 $\begin{matrix} q_{1}^{1} \end{matrix}$ 和 $\begin{matrix} q_{1}^{2} \end{matrix}$ , 即在任意时刻均满足 $\begin{matrix} q_{1}^{*} = q_{1}^{1} + q_{1}^{2} \end{matrix}$ , 假设 $\begin{matrix} T = T_{1}^{*} \end{matrix}$ 和 $\begin{matrix} T = T_{2}^{*} \end{matrix}$ 分别为 $\begin{matrix} q_{1} = q_{1}^{1} \end{matrix}$ 和 $\begin{matrix} q_{1} = q_{1}^{2} \end{matrix}$ 时满足式(6), (8), (10)和(11)的解, 令 $\begin{matrix} T_{3}^{*} = T_{1}^{*} + T_{2}^{*} - T_{0} \end{matrix}$ , 则 $\begin{matrix} T_{3}^{*} \end{matrix}$ 满足:

(12)	$\begin{matrix} λ \frac{? T}{? x} \|_{x = 0} = - q_{1}^{1} (τ) - q_{1}^{2} (τ) = - q_{1}^{*} (τ) \end{matrix}$

(13)	$\begin{matrix} T (x, 0, q_{1}) = T_{0} + T_{0} - T_{0} = T_{0} \end{matrix}$

(14)	$\begin{matrix} λ \frac{? T}{? x} \|_{x = d} = 0 + 0 = 0 \end{matrix}$

由于式(8)中 $\begin{matrix} ρ, c \end{matrix}$ 和 $\begin{matrix} λ \end{matrix}$ 作为与温度、时间无关的常数处理, 故 $\begin{matrix} T_{3}^{*} \end{matrix}$ 也满足式(8). 式(12)即为 $\begin{matrix} q_{1} = q_{1}^{*} \end{matrix}$ 时的式(6), 而式(13)和(14)与式(10)和(11)相同, 即 $\begin{matrix} T_{3}^{*} \end{matrix}$ 也为 $\begin{matrix} q_{1} = q_{1}^{*} \end{matrix}$ 时满足式(6), (8), (10)和(11)的解, 由解的唯一性知 $\begin{matrix} T_{3}^{*} = T^{*} \end{matrix}$ . 由以上可知, 当 $\begin{matrix} q_{1}^{*} = q_{1}^{1} + q_{1}^{2} \end{matrix}$ 时, $\begin{matrix} q_{1}^{*} \end{matrix}$ 对温度场造成的影响 $\begin{matrix} (T^{*} - T_{0}) \end{matrix}$ 也相当于 $\begin{matrix} q_{1}^{1} \end{matrix}$ 和 $\begin{matrix} q_{1}^{2} \end{matrix}$ 对温度场造成的影响 $\begin{matrix} (T_{1}^{*} - T_{0}) \end{matrix}$ 和 $\begin{matrix} (T_{2}^{*} - T_{0}) \end{matrix}$ 的叠加, 同理, 将 $\begin{matrix} q_{1}^{*} \end{matrix}$ 看做M股(M为任意正整数)热流的叠加 $\begin{matrix} \sum_{m = 1}^{M} q_{1}^{m} \end{matrix}$ 时, 即:

(15)	$\begin{matrix} q_{1}^{*} = \sum_{m = 1}^{M} q_{1}^{m} \end{matrix}$

若 $\begin{matrix} T = T_{m}^{*} \end{matrix}$ 是 $\begin{matrix} q_{1} = q_{1}^{m} \end{matrix}$ 时满足式(6), (8), (10)和(11)的解, 则有:

(16)	$\begin{matrix} T^{} - T_{0} = \sum_{m = 1}^{M} (T_{m}^{} - T_{0}) \end{matrix}$

亦即若将 $\begin{matrix} q_{1}^{*} \end{matrix}$ 看做M股热流的叠加 $\begin{matrix} \sum_{m = 1}^{M} q_{1}^{m} \end{matrix}$ , 则这些热流对温度场的影响相互独立, 总的效果为各热流的效果的简单叠加. 同样的, 对于 $\begin{matrix} q_{2} \end{matrix}$ 而言, 若将 $\begin{matrix} q_{2} \end{matrix}$ 看做若干股热流的叠加, 则这些热流对温度场的影响也相当于各热流的影响的简单叠加, 而且 $\begin{matrix} q_{1} 和 q_{2} \end{matrix}$ 对于型壳温度场的影响相当于 $\begin{matrix} q_{1} 和 q_{2} \end{matrix}$ 各自对温度场的影响的简单叠加.

1.3 界面换热系数求解方法

界面换热系数的计算采用Beck等^[25]提出的非线性估算法, 其基本原理是: 在某一时刻 t^*之前界面换热系数已经计算完毕(已知); 假设从这一时刻起N个时间步长Nd t内界面换热系数h保持不变, 其值为 $\begin{matrix} h^{*} \end{matrix}$ , 基于之前的界面换热系数和温度场, 以及此时的界面换热系数 $\begin{matrix} h^{*} \end{matrix}$ 计算未来N个时间步长内温度场的变化, 即:

(17)	$\begin{matrix} h_{τ^{}} = h_{τ^{} + d τ} = \dots = h_{τ^{} + N d τ} = h^{} \end{matrix}$

(18)	$\begin{matrix} T = T (x, τ^{*} + d τ, q_{1}, q_{2}) \end{matrix}$

式(18)中 $\begin{matrix} q_{1} = h^{*} (T_{c a s t} - T |_{x = 0^{+}}), \end{matrix}$ $\begin{matrix} q_{2} = σ ε (T^{4} |_{x = d^{-}} - T_{B}^{4}) \end{matrix}$ (其中, $\begin{matrix} T_{c a s t} \end{matrix}$ 为计算得到的铸件表面温度; $\begin{matrix} σ \end{matrix}$ 和 $\begin{matrix} ε \end{matrix}$ 为有效辐射率和黑体辐射常数; $\begin{matrix} T_{B} \end{matrix}$ 为背景温度, 即真空炉内壁温度).

定义 $\begin{matrix} f (h^{*}) \end{matrix}$ 为:

(19)	$\begin{matrix} f (h^{}) = \sum_{n = 1}^{N} \sum_{k = 1}^{K} (T (x_{k}, τ^{} + n d τ, q_{1}, q_{2}) - T_{x_{k}, τ^{*} + n d τ}^{m})^{2} \end{matrix}$

式(19)中, $\begin{matrix} T_{x_{k}, τ^{*} + n d τ}^{m} \end{matrix}$ 为 $\begin{matrix} τ^{*} + n d τ \end{matrix}$ 时刻 $\begin{matrix} x_{k} \end{matrix}$ 点测得的温度. $\begin{matrix} f (h^{*}) \end{matrix}$ 代表 t^*时刻之后界面换热系数保持为 $\begin{matrix} h^{*} \end{matrix}$ 的情况下计算得到的温度场与测量所知的温度场之间的差异. $\begin{matrix} f (h^{*}) \end{matrix}$ 越小, 则表明对 t^*时刻之后的界面换热系数估计值h越接近真实情况. 因此计算过程以极小化 $\begin{matrix} f (h^{*}) \end{matrix}$ 为目的, 当 $\begin{matrix} f (h^{*}) \end{matrix}$ 达到极小值时的 $\begin{matrix} h^{*} \end{matrix}$ 即为 t^*时刻所求的界面换热系数.

2 实验设计

本工作采用了左右对称的2块板状铸件, 以利于获取较多实验数据. 实验采用的模型及热电偶位置安排如图2所示. 铸件板状部分尺寸为170 mm×170 mm×25 mm. 不同热电偶分别埋入铸件或型壳中. 埋入铸件的热电偶, 编号从上到下依次为TC casting 1, 2, 3和4, 分别位于冒口、板状铸件中央、内浇道和直浇道中, 其中TC casting 1, 2和4号热电偶采用钨铼热电偶, 用来测量铸件温度变化, TC casting 3号热电偶为K型热电偶, 用来测量金属液到达时间. 埋入型壳中的热电偶处于同一水平高度, 横向间距10 mm, 位于板状铸件中央附近, 热电偶顶端到铸件/型壳界面距离依次为3, 6和9 mm, 按照离界面由近到远其编号依次为TC mold 1, 2和3, 均为K型热电偶. 铸件本身为对称造型, 在对侧于同样位置放置同样的热电偶, 可在一次浇注中获得2组数据用于反算.

	Figure Option View Download New Window
	Fig.2 Schematic of the cavity and the positions of the thermocouples (TC)图2 铸型型腔形状及热电偶布置示意图

实验采用的型壳材料热物性通过实验测得. 型壳材料比热容随温度的变化如图3所示, 其在900 ℃左右的突然上升与型壳中高岭土成分的莫来石化有关. 型壳材料导热系数受温度影响很小, 基本保持在1.4 W/(mK)左右, 密度则为2170 kg/m³. 实验采用的Ti6Al4V合金名义成分(质量分数)为: Al 6%, V 4%, 余量为Ti, 其比热容及导热系数随温度的变化见图4. Ti6Al4V比热容和导热系数在1620~1650 ℃的突变对应于熔化/凝固过程, 而980~1000 ℃的突变对应于固态相变.

	Figure Option View Download New Window
	Fig.3 Variation of specific heat capacity (c) of shell with temperatures (T)图3 型壳材料比热容(c)随温度的变化

	Figure Option View Download New Window
	Fig.4 Variations of c (a) and thermal conductivity l (b) of Ti6Al4V with temperatures图4 Ti6Al4V合金的c和导热系数 l随温度的变化

3 结果与讨论

3.1 实验测温及反求结果

图5是实验测得的一侧热电偶的温度. 根据多次测量的数据, 热电偶不带保护套时浇注后数秒内温度升至1700 ℃左右, 而带保护套时升温较慢, 最高温度在约1570 ℃处停留较长时间, 而Ti6Al4V合金两相区为1620~1650 ℃, 取其中值1635 ℃, 估计出2号热电偶处测量得到的温度大约比实际温度低4%左右. 而型壳中的热电偶工作端均随同型壳一起烧结, 与型壳接触较好, 因此认为测得的温度即为相应位置型壳温度. 图6是根据TC casting 2和TC mold 1~3号热电偶的温度反求得到的h- t关系图, 其中插图是对h- t关系在0~200 s内变化情况的局部放大.

	Figure Option View Download New Window
	Fig.5 Measured temperatures at different positions in mold and casting ( t—time)图5 实验测得铸型及铸件各处的温度

	Figure Option View Download New Window
	Fig.6 Calculated interfacial heat transfer coefficient (h) between 0~4000 s and between 0~200 s (inset)图6 反求得到的界面换热系数(h)及局部放大图

3.2 界面换热系数反求精度评估

3.2.1铸型材料比热容、热扩散率及热电偶定位误差对计算结果影响评估在一维传热模型中, 若铸件/铸型之间h随铸件状态的变化关系已知, 铸件、铸型的热物性参数准确且初始状态已知, 各热电偶定位准确, 测得的温度无误差, 通过温度场模拟可以得到各热电偶处的温度变化. 根据模拟得到的各处温度变化, 结合铸件、铸型的热物性参数和初始状态, 各热电偶的定位位置, 可以通过反算精确地得到h的变化. 然而实际情况下铸型的热物性参数及热电偶的位置并不总是非常准确, 而O'Mahoney等^[9]指出, 这些误差对反算结果会造成很大的影响, 因此有必要评估这些误差造成了多大的影响并尽量消除它们的影响.

图7~9分别是当计算采用的铸型材料c、热扩散率 a和热电偶实际位置偏离预定位置时计算得到的h与预设h之间的关系以及在条件A, C, E下反算过程中得到的铸件、铸型温度与此前无误差情况下模拟得到的温度的对比. 图中条件A~F依次为: 计算采用的c偏小和偏大20%, 计算采用的 a偏小和偏大20%, 以及实际的热电偶位置为2.5, 5.5, 8.5 mm和3.5, 6.5, 9.5 mm.

	Figure Option View Download New Window
	Fig.7 Calculated h based on conditions A and B (when assumed c is 20% smaller and bigger than the real c) (a) and temperatures based on condition A (b)图7 假定c发生偏差时计算得到的h与模拟温度

	Figure Option View Download New Window
	Fig.8 Calculated h based on conditions C and D (when assumed the thermal diffusivity of the shell deviate ( a) is 20% smaller and bigger than the real a) (a) and temperatures based on condition C (b)图8 假定铸型材料热扩散率( a)发生偏差时计算得到的h与模拟温度

	Figure Option View Download New Window
	Fig.9 Calculated h based on conditions E and F (when real distances between the interface and thermocouples are 2.5, 5.5, 8.5 mm, and 3.5, 6.5, 9.5 mm) (a) and temperatures based on condition E (b)图9 热电偶埋入型壳位置发生偏离时反算得到的h与模拟温度

对比图7~9表明, 铸型材料c, a和热电偶实际位置偏离预定位置带来的h的误差和反算得到的温度变化误差有明显不同的特点: c的误差对铸件温度和铸型中各点的温差有明显的影响, 而且会对h的计算造成很大的误差; a的误差对铸件温度影响不大, 而对铸型中各点温差有明显的影响, 对h的计算精度影响不大; 热电偶定位误差对铸件温度和h的计算精度的影响几乎完全体现在浇注开始后约50 s内, 对后续反算的影响也完全继承自最初50 s内的影响, 同时热电偶定位误差对铸型内各点的温差基本没有影响.

3.2.2 误差评估与校正基于对型壳材料比热容和导热系数的测量结果, 可认为c≈1290 J/(kgK), l≈1.4 W/(mK), a≈5.0×10^-⁷m²/s. 实验中型壳平均厚度约为18 mm, 热电偶与界面间的等效最小距离为3.5 mm, 浇注温度约为1720 ℃(考虑到Ti6Al4V合金液进入型腔之前在直浇道和横浇道中与铸型接触, 同时本实验采用重力浇注, 流速较慢, 认为Ti6Al4V合金进入型腔后的平均温度约为1680 ℃).

图10为0~750 s内h反求过程中计算得到的各测温点温度与实测温度的对比. 模拟得到的型壳最高温度约为840 ℃, 可以认为型壳中高岭土成分的莫来石化影响很小. 从图10可以看出, 按照原有的型壳热物性参数计算得到的型壳中不同位置热电偶的温差明显小于实测不同热电偶的温差, 铸件温度下降速率明显低于实际值, 从之前的评估可以认为采用的型壳 a高于实际值, 而c低于实际情况. 此外, 铸件温度开始明显下降的模拟时间明显低于实际时间, 表明热电偶到界面的最小等效距离(与距界面最近的热电偶温度相等的型壳内部温度点到界面的距离)低于实际值.

	Figure Option View Download New Window
	Fig.10 Comparison between measured and calculated temperatures图10 反求得到的各测温点温度与实测温度对比

针对上述问题, 将c适当上调, 将 a适当下调, 同时微调热电偶到界面的最小等效距离以保证反算得到的铸件温度与实测值相符以及铸型中不同测温点温差与实测温差相符. 经多次调整后, 当c≈1410 J/(kgK), a≈4.09×10^-⁷m²/s, 热电偶到界面的最小等效距离为3.6 mm时, 反算得到的铸件铸型温度与实测值吻合较好, 图11为修正后反算得到的h及铸件铸型的温度.

	Figure Option View Download New Window
	Fig.11 Calculated h between 0~4000 s and between 0~200 s (inset) (a), comparison between measured and calculated temperatures with specific heat capacity, thermal diffusivity of shell and positions of thermocouples adjusted (b)图11 修正铸型材料c, a和热电偶定位位置之后反求得到的h和各测温点温度与实测温度对比

从图11可以看出, 修正铸型材料热物性参数及热电偶定位位置后反算过程中得到的铸件及铸型温度与实测温度吻合良好. 从反算得到的h随时间变化的关系来看, 浇注开始后约20 s内h变化剧烈, 从0上升至约440 W/(m²K), 很快又下降至约184 W/(m²K), 对应于金属液开始与铸型接触并于表面生成一层凝壳. 随后随着凝固的进行, h缓慢下降至约110 W/(m²K), 这一阶段随着凝壳的增厚和铸件的收缩, 凝壳与铸型间的接触状况不断恶化, h的变化量相当于峰值的20%. 铸件完全凝固后, h随着铸件的冷却而降低, 这一阶段由于铸件不再有明显的收缩, h的下降速度较前一阶段还要低得多.

3.3 界面换热系数分析

3.3.1 界面换热系数与铸件状态的关系如上节所述, 浇注后, 铸件状态的变化经过以下几个阶段: 完全液态, 表面生成凝固壳层, 凝固层厚度不断增大, 铸件完全凝固. 反算不仅可以得到h随时间的变化, 也可以得到铸件内部的温度、凝固分数分布. 在铸件完全凝固前, 以温度低于Ti6Al4V合金液相线的部分为凝固层, 则可以得到铸件表面凝固层厚度 d随时间的变化.

基于h和 d随时间的变化, 可以得到这2者之间的关系, 如图12所示. 图12表明, 铸件表面形成凝固层前, h仍处于峰值附近, 而凝固层一旦形成, 则h急剧下降. 当凝固层厚度约为0.8 mm时, h降至约184 W/(m²K), 随后随着凝固层厚度的增加而缓慢下降. 当凝固层厚度达到11 mm时, h与完全凝固时相差无几.

	Figure Option View Download New Window
	Fig.12 Relationship between h and the thickness of solidified layer d图12 h随铸件表面凝固层厚度 d变化的关系

因此, 为了简化起见,可以将铸件完全凝固前h随铸件固相分数变化划分为以下3段:

(20)	$\begin{matrix} h = \{\begin{matrix} 440 (δ = 0) \\ - 319.96 δ + 44 0 (δ < 0.8 m m) \\ - 8.1523 δ + 190.6 4 (0.8 m m < δ < 11 m m) \end{matrix} \end{matrix}$

其中第1段中, 金属液未凝固, h有波动但影响不大. 第2段中, 金属液与铸型的表面开始形成凝壳, 糊状区向铸件内部推进. 由于完整的凝壳逐渐形成, 铸件与型壳的接触状态开始发生改变, 因此h急剧减小, 降幅近60%. 第3段中, 铸件表层已经形成完整的凝壳, 其与型壳的接触状态固定下来, 不再发生明显改变, 因此h降低的速率大大减慢, 这一阶段h的降幅约为其峰值的20%. 此后, 铸件基本凝固, 温度开始快速下降, 此后h主要取决于界面两侧的温度和辐射率等.

铸件凝固后, 基于h和铸件表面温度T随时间的变化, 可以得到它们之间的关系, 如图13所示. 图13表明, 铸件基本凝固之后, 其与型壳间的h随铸件温度下降, 其间虽有波动, 但影响不大. 铸件凝固后h与铸件温度间的关系可简化为:

$\begin{matrix} h = 4.2584 \times 10^{- 8} T^{3} - 6.5324 \times 10^{- 5} T^{2} + \end{matrix}$

(21)	$\begin{matrix} 5.0115 \times 10^{- 2} T + 21.478 \end{matrix}$

	Figure Option View Download New Window
	Fig.13 Relationship between h and T of the casting after solidification图13 铸件基本凝固后h随铸件温度变化的关系

3.3.2 界面换热系数的验证将式(20)和(21)代入图2所示的铸件的三维模型中, 即铸件表面每一处的h都根据表面单元及附近单元的凝固状态或表面单元的温度来决定, 其它热物性参数等均与一维反算模型相同, 可得到铸件及铸型内3根热电偶处的温度变化. 将通过以上方法计算得到的温度与实验测得的温度相比, 可得到图14. 从图14可以看出, 根据反求得到的h计算出的型壳和铸件温度与测量到的温度基本相符, 表明反求得到的h具有较好的准确度, 也证明选取的计算参数具有较高的准确度. 同时, 计算得到的温度与测量到的温度间有部分位置有一定差异, 表明本反求模型仍然存在一定的误差. 造成这些误差的原因主要有: (1) 实验中铸件-铸型系统并非标准的一维模型, 从而使反算存在系统性误差; (2) 型壳与铸型本身形状的不精确, 以及布置的热电偶对温度场的影响, 造成温度偏离实际情况; (3) 型壳、铸件热物性参数及热电偶位置等实验参数的偏差, 从而导致计算与实际不一致.

	Figure Option View Download New Window
	Fig.14 Simulated temperatures of casting and shell based on the fitted h图14 根据反求得到的h 计算出的型壳和铸件温度

4 结论

(1) 建立了熔模铸造铸件/型壳换热一维传热数学模型, 讨论了在导热系数和比热容不受温度影响的情况下铸型内部温度场与铸型内壁热流密度之间的关系, 从数学上分析了比热容、导热系数、热电偶定位和热电偶测温误差等对界面热流和界面换热系数反算的影响和它们之间的不同.

(2) 从数值模拟的角度评估了比热容、导热系数和热电偶定位误差对铸件、铸型各处温度的影响大小, 结合初步反求结果对各项误差进行了校正.

(3) 获得了较为准确的界面换热系数, 分析了界面换热系数与铸件状态(凝固层厚度、温度)之间的关系. 分析表明, 在铸件表面形成凝固层期间, 界面换热系数急剧下降约60%; 凝固层厚度增长期间, 界面换热系数下降速度放缓, 降幅接近峰值的20%; 其后铸件已凝固, 铸件温度下降迅速, 界面换热系数随同铸件温度一起下降, 但下降速度比前两个阶段小得多.

(4) 在三维铸件/铸型模型中对这一关系进行了验证. 验证结果表明, 校正后的计算参数基本准确, 反算得到的界面换热系数随铸件状态变化的关系较为可靠.

参考文献

View Option

[1]	Liu J, Yang Y, Lu D. Foundry, 2008;57: 1155 (刘剑, 杨屹, 卢东. 铸造, 2008;57: 1155) [本文引用:2]
[2]	Atwood R C, Lee P D, Curtis R V, Maijer D M. Dent Mater, 2007;23(1): 60 [本文引用:1] [JCR: 4.16]
[3]	Liang Z J, Xu Q Y, Li J T, Li S Q, Zhang J, Liu B C, Zhong Z Y. Rare Met Mater Eng, 2003;32: 164 (梁作俭, 许庆彦, 李俊涛, 李世琼, 张继, 柳百成, 仲增墉. 稀有金属材料与工程, 2003;32: 164) [本文引用:1]
[4]	Guo Z P, Xiong S M, Cho S H, Choi J K. Acta Metall Sin, 2007;43: 1149 (郭志鹏, 熊守美, 曹尚铉, 崔正吉. 金属学报, 2007;43: 1149) [本文引用:2] [JCR: 0.548] [CJCR: 0.835]
[5]	Guo Z P, Xiong S M, Cho S H, Choi J K. Acta Metall Sin, 2007;43: 1155 (郭志鹏, 熊守美, 曹尚铉, 崔正吉. 金属学报, 2007;43: 1155) [本文引用:1] [JCR: 0.548] [CJCR: 0.835]
[6]	Cao Y Y, Guo Z P, Xiong S M. China Foundry, 2014;11: 314 [本文引用:1] [JCR: 0.341] [CJCR: 0.1885]
[7]	Guo Z P, Xiong S M, Liu B C, Mei L, Allison J. Int J Heat Mass Transfer, 2008;51: 6032 [本文引用:1] [JCR: 2.522]
[8]	Long A, Thornhill D, Armstrong C, Watson D. Appl Therm Eng, 2011;31: 3996 [本文引用:1] [JCR: 2.624]
[9]	O'Mahoney D, Browne D J. Exp Therm Fluid Sci, 2000;22: 111 [本文引用:2] [JCR: 2.08]
[10]	Yuan S, Bu K, Dong Y W. Foundry Technol, 2012;33: 177 (袁帅, 卜昆, 董一巍. 铸造技术, 2012;33: 177) [本文引用:1] [CJCR: 0.544]
[11]	Konrad C H, Brunner M, Kyrgyzbaev K, V?lkl R, Glatzel U. J Mater Process Technol, 2011;211: 181 [本文引用:1]
[12]	Zhang W H, Xie G N, Zhang D. Exp Therm Fluid Sci, 2010;34: 1068 [本文引用:1] [JCR: 2.08]
[13]	Dong Y W, Bu K, Dou Y Q, Zhang D H. J Mater Process Technol, 2011;211: 2123 [本文引用:1]
[14]	Rafique M M A, Iqbal J. Int J Heat Mass Transfer, 2009;52: 2132 [本文引用:1] [JCR: 2.522]
[15]	Jin H P, Li J R, Yu J, Liu S Z. Rare Met Mater Eng, 2010;39: 767 [本文引用:1]
[16]	Sun Z Z, Hu H, Niu X P. J Mater Process Technol, 2011;211: 1432 [本文引用:2]
[17]	Ilkhchy A F, Jabbari M, Davami P. Int Commun Heat Mass Transfer, 2012;39: 705 [本文引用:1] [JCR: 2.124]
[18]	Aweda J O, Adeyemi M B. J Mater Process Technol, 2009;209: 1477 [本文引用:1]
[19]	Xu H, Hou H, Zhao Y H, Chen Z. Chin J Nonferrous Met, 2003;13: 881 (徐宏, 侯华, 赵宇宏, 陈铮. 中国有色金属学报, 2003;13: 881) [本文引用:2]
[20]	Palumbo G, Piglionico V, Piccininni A, Guglielmi P, Sorgente D, Tricarico L. Appl Therm Eng, 2015;78: 682 [本文引用:1] [JCR: 2.624]
[21]	Chen L P, Wang Y X, Peng L M, Fu P H, Jiang H Y. Exp Therm Fluid Sci, 2014;54: 196 [本文引用:1] [JCR: 2.08]
[22]	Wang Z F, Yao M, Wang X D, Zhang X B, Yang L S, Lu H Z, Wang X. J Mater Process Technol, 2014;214: 44 [本文引用:2]
[23]	Watanabe K, Miyakawa O, Takada Y, Okuno O, Okabe T. Biomaterials, 2003;24: 1737 [本文引用:1] [JCR: 8.312]
[24]	Liang Z J, Xu Q Y, Li J T, Li S Q, Zhang J, Liu B C, Zhong Z Y. Rare Met Mater Eng, 2002;31: 353 (梁作俭, 许庆彦, 李俊涛, 李世琼, 张继, 柳百成, 仲增墉. 稀有金属材料与工程, 2002;31: 353) [本文引用:1]
[25]	Beck J V, Blackwell B, St Clair C R. Inverse Heat Conduction: Ill-Posed Problems. New York: Wiley, 1985: 145 [本文引用:1]

2008

0.0

... 在熔模铸造工艺开发过程中, 数值模拟因为能够预测缺陷的形成, 减少工艺开发所需的时间和成本而受到青睐, 得到越来越多的应用^[1-3] ...

... 通常而言, 前者可以通过查阅文献或实验测量得到较为准确的数值, 后者都是通过估计给出, 且经常作为常数使用^[1,2], 可靠性和精确性都有待商榷 ...

2007

4.16

0.0

Atwood

R C

, Lee

P D

, Curtis

R V

, Maijer

D M

. Dent Mater, 2007;23(1): 60

Abstract

Objective

The objective of this study was to apply computational modeling tools to assist in the design of titanium dental castings. The tools developed should incorporate state-of-the-art micromodels to predict the depth to which the mechanical properties of the crown are affected by contamination from the mold. The model should also be validated by comparison of macro- and micro-defects found in a typical investment cast titanium tooth crown.

Methods

Crowns were hand-waxed and investment cast in commercial purity grade 1 (CP-1) titanium by a commercial dental laboratory. The castings were analyzed using X-ray microtomography (XMT). Following sectioning, analysis continued with optical and scanning electron microscopy, and microhardness testing.

An in-house cellular-automata solidification and finite-difference diffusion program was coupled with a commercial casting program to model the investment casting process. A three-dimensional (3D) digital image generated by X-ray tomography was used to generate an accurate geometric representation of a molar crown casting. Previously reported work was significantly expanded upon by including transport of dissolved oxygen and impurity sources upon the arbitrarily shaped surface of the crown, and improved coupling of micro- and macro-scale simulations.

Results

Macroscale modeling was found to be sufficient to accurately predict the location of the large internal porosity. These are shrinkage pores located in the thick sections of the cusp. The model was used to determine the influence of sprue design on the size and location of these pores. Combining microscale with macroscale modeling allowed the microstructure and depth of contamination to be predicted qualitatively. This combined model predicted a surprising result—the dissolution of silicon from the mold into the molten titanium is sufficient to depress the freezing point of the liquid metal such that the crown solidifies the subsurface. Solidification then progresses inwards and back out to the surface through the silicon-enriched near-surface layer. The microstructure and compositional analysis of the near-surface region are consistent with this prediction.

Significance

A multiscale model was developed and validated, which can be used to design CP-Ti dental castings to minimize both macro- and micro-defects, including shrinkage porosity, grain size and the extent of surface contamination due to reaction with the mold material.

The model predicted the surprising result that the extent of Si contamination from the mold was sufficient to suppress the liquidus temperature to the extent that the surface (to a depth of ∼100 μm) of the casting solidifies after the bulk. This significantly increases the oxygen pickup, thereby increasing the depth of formation of alpha casing. The trend towards mold materials with reduced Si in order to produce easier-to-finish titanium castings is a correct approach.

... 通常而言, 前者可以通过查阅文献或实验测量得到较为准确的数值, 后者都是通过估计给出, 且经常作为常数使用^[1,2], 可靠性和精确性都有待商榷 ...

2003

0.0

... 在熔模铸造工艺开发过程中, 数值模拟因为能够预测缺陷的形成, 减少工艺开发所需的时间和成本而受到青睐, 得到越来越多的应用^[1-3] ...

2007

0.548

0.835

Guo

Z P

, Xiong

S M

, Cho

S H

, Choi

J K

. Acta Metall Sin, 2007;43: 1149

(郭志鹏, 熊守美, 曹尚铉, 崔正吉. 金属学报, 2007;43: 1149)

采用“阶梯”模具，设计了压铸过程模具温度测量的实验方案并进行了压铸实验，以实验中测得的铸型内部不同位置的温度为基础，采用热传导反算法求解了压铸过程中铸件/铸型界面热流以及换热系数。分析了铸件的厚度对于界面热流以及换热系数的影响，研究结果表明：压铸过程铸件/铸型界面热流或是换热系数随着压射过程的进行迅速升高直至最大值，然后随着凝固过程的进行而减小。同时，铸件的不同厚度部位与铸型之间的界面热流和换热系数的变化规律也不同，随着铸件厚度的增大，件/型之间的界面热流和换热系数峰值都要减小，但是界面热流和换热系数的较大值保持的时间则逐渐增大。

... 近年来, 国内外学者对铸造过程铸件/铸型界面换热系数的研究, 主要涉及压铸^[4-8]、熔模铸造^[9-15]、挤压铸造^[16-18]、砂型铸造^[19,20]和连续铸造^[21,22]等, 包含铝合金、镁合金、高温合金和钢等, 取得了较好的效果 ...

... 郭志鹏等^[4-7]针对铝合金压铸条件下的界面换热行为展开了大量的研究, 建立了完善的数值模型, 分析了铸件厚度、压铸压力、铸型温度和测量位置等对界面换热系数的影响 ...

2007

0.548

0.835

Guo

Z P

, Xiong

S M

, Cho

S H

, Choi

J K

. Acta Metall Sin, 2007;43: 1155

(郭志鹏, 熊守美, 曹尚铉, 崔正吉. 金属学报, 2007;43: 1155)

通过采用本文第1部分提出的数学模型，本文的第2部分将求解各种工艺参数下铸件/铸型间的界面热流和换热系数，重点研究不同工艺参数如铸造压力、低速速度、高速速度、浇铸温度和铸型温度等对于界面热流和换热系数的影响。在现有的“阶梯”块铸件的条件下，计算结果表明：压铸过程各种工艺参数对于铸件/铸型界面热流和换热系数有着不同的影响规律。铸型初始的模腔表面温度对于界面热流的峰值有着很大的影响，而且随着铸型初始模腔表面温度的上升，热流峰值在不断地下降。其它工艺参数对于热流峰值则没有很大的影响作用。对于较厚的 “阶梯”面，铸型初始的模腔表面温度对于界面换热系数的影响较大，与对热流的影响相似，随着该温度的上升，界面换热系数的峰值也在不断地下降。而其他工艺参数则对其影响不大。对于较薄的 “阶梯”面，各种工艺参数对于界面换热系数的影响不大。

2014

0.341

0.1885

2008

2.522

0.0

2011

2.624

0.0

2000

2.08

0.0

... 然而实际情况下铸型的热物性参数及热电偶的位置并不总是非常准确, 而O'Mahoney等^[9]指出, 这些误差对反算结果会造成很大的影响, 因此有必要评估这些误差造成了多大的影响并尽量消除它们的影响 ...

2012

0.0

0.544

2011

0.0

... Konrad等^[11]针对IN738LC高温合金熔模铸造条件下的界面换热行为, 测量了陶瓷型壳的比热容和热扩散率, 研究了型壳预热温度和厚度对铸件晶粒尺寸和二次臂间距的影响, 并反算得到各工艺参数(型壳预热温度和型壳厚度)下铸件凝固过程中界面换热系数随铸件温度的变化 ...

2010

2.08

0.0

Zhang

W H

, Xie

G N

, Zhang

. Exp Therm Fluid Sci, 2010;34: 1068

Abstract

In order to effectively improve the numerical prediction accuracy in a blade investment casting process, a new method is proposed to determine the interfacial heat transfer coefficient (IHTC) in a complicated blade casting by combining the numerical prediction, optimization and limited experimental data. An investment experiment of the blade is conducted to acquire the surface temperature of the casting and the shell mould. Regarding the complicated mechanism of the interfacial heat transfer in the progressive solidification, a new continuous model with three-step evolution is established for the casting–mould IHTC, and a power function is proposed to correlate the mould–environment IHTC with solidification time as well. A globally convergent method is employed to search the optimal coefficients involved in the IHTCs correlations. Results show that the predicted temperature based on proposed models agrees well with the experimental data with the maximum deviation being less than 5.5%, and a significant variation of the casting–mould IHTC is observed. It is concluded that the prediction accuracy and efficiency associated with the optimization method can be greatly improved with the present IHTC models.

2011

0.0

2009

2.522

0.0

2010

0.0

2011

0.0

... Sun等^[16]研究了挤压铸造条件下AM60 镁合金与金属铸型之间的界面换热行为, 比较多项式拟合方法与反算法计算得到的界面换热系数, 结果表明反算法的精度较高 ...

2012

2.124

0.0

2009

0.0

2003

0.0

... 徐宏等^[19]研究了砂型铸造条件下ZL104与铸型间的界面换热系数, 给出了界面换热系数随时间的变化 ...

2015

2.624

0.0

2014

2.08

0.0

2014

0.0

... Wang等^[22]测量了连续铸造条件下到弯月面不同距离处的钢材表面温度, 反算得到连续铸造过程中的界面换热系数, 并以此预测不同位置钢材表面凝壳厚度, 预测结果与实测结果吻合良好 ...

2003

8.312

0.0

Watanabe

, Miyakawa

, Takada

, Okuno

, Okabe

. Biomaterials, 2003;24: 1737

Abstract

Since dental casting requires replication of complex shapes with great accuracy, this study examined how well some commercial titanium alloys and experimental titanium–copper alloys filled a mold cavity. The metals examined were three types of commercial dental titanium [commercially pure titanium (hereinafter noted as CP-Ti), Ti–6Al–4V (T64) and Ti–6Al–7Nb (T67)], and experimental titanium–copper alloys [3%, 5% and 10% Cu (mass %)]. The volume percentage filling the cavity was evaluated in castings prepared in a very thin perforated sheet pattern and cast in a centrifugal casting machine. The flow behavior of the molten metal was also examined using a so-called “tracer element technique.” The amounts of CP-Ti and all the Ti–Cu alloys filling the cavity were similar; less T64 and T67 filled the cavity. However, the Ti–Cu alloys failed to reach the end of the cavities due to a lower fluidity compared to the other metals. A mold prepared with specially designed perforated sheets was effective at differentiating the flow behavior of the metals tested. The present technique also revealed that the more viscous Ti–Cu alloys with a wide freezing range failed to sequentially flow to the end of the cavity.

... 钛合金熔点高, 化学活性高, 多采用熔模铸造工艺, 铸型采用陶瓷材料, 铸造过程在真空炉中完成^[23,24], 大大增加了实验设计与数据处理的难度, 目前尚鲜有对其铸造过程铸件/铸型界面换热系数的研究, 无法为生产过程提供更多的帮助 ...

2002

0.0

1985

0.0

... 3 界面换热系数求解方法界面换热系数的计算采用Beck等^[25]提出的非线性估算法, 其基本原理是: 在某一时刻t^*之前界面换热系数已经计算完毕(已知) ...