板带热轧中间坯厚度分布规律研究

板带热轧中间坯厚度分布规律研究 Research on thickness distribution rule of intermediate slab during hot strip rolling corresp first-author 彭文，博士，副研究员，从事轧制过程自动控制、模型优化、故障诊断等研究；E-mail：pengwen233@163.com 彭文，博士，副研究员，从事轧制过程自动控制、模型优化、故障诊断等研究；E-mail：pengwen233@163.com pengwen233@163.com 东北大学 轧制技术及连轧自动化国家重点实验室，辽宁 沈阳，110819 东北大学 轧制技术及连轧自动化国家重点实验室，辽宁 沈阳，110819 State Key Laboratory of Rolling and Automation, Northeastern University, Shenyang 110819, China State Key Laboratory of Rolling and Automation, Northeastern University, Shenyang 110819, China 伍锦花 在热连轧生产过程中，受测量环境和测量手段的限制，尚未实现对中间坯厚度的实时在线精确测量，对产品最终的厚度精度和精轧区穿带过程稳定性造成影响。为解决此问题，采用轧制特性法分析中间坯厚度产生偏差的原因；为进一步研究各因素对厚度偏差影响的规律，通过影响系数法计算中间坯长度方向上温度和厚度分布，建立热力耦合有限元模型，并将有限元结果与计算结果进行比较。研究结果表明：随着轧制过程的进行，在轧制方向上，中间坯温度逐渐降低，头、尾温度的平均偏差约为22 ℃，头、尾最大温度偏差达到32 ℃；由于温度偏差导致中间坯厚度呈变大趋势，中间坯头尾厚度偏差为0.20 mm，长度方向最大厚度偏差达0.53 mm；本文得到的中间坯厚度和温度的分布能够为后续精轧过程提供前提条件，为高精度控制成品轧件的厚度以及维持穿带稳定性提供依据。 Due to the restriction of environment and the measuring methods, the real-time online accurate measurement of the thickness of intermediate billet has not been realized, which affects the final thickness accuracy and the stability of strip penetration in finishing zone. To solve this problem, the reason for thickness deviation of intermediate billet was discussed by means of rolling characteristic analysis during hot rolling. In order to describe the influence of various factors on the thickness deviation, the temperature and thickness distribution along the length direction of the intermediate billet was calculated by the influence coefficient method, a three-dimensional dynamic thermo-mechanical coupling finite element model was established, and the finite element results were compared with the calculated results. The results show that as the rolling process progresses, the temperature of intermediate billet decreases gradually along the rolling length direction, and the average deviation of the temperature of head and tail is about 22 ℃, and the maximum temperature deviation of head and tail is 32 ℃. Due to the temperature deviation, the thickness of the intermediate billet tends to be thicker. The thickness deviation of the intermediate billet head and tail is 0.200 mm, and the maximum thickness deviation of the length direction is 0.53 mm. The obtained thickness and temperature distribution of the intermediate billet can provide the premise for the subsequent fine rolling process and provide basis for the high precision thickness control and the stability of the strip penetration. 热连轧 轧制特性分析 有限元法 厚度分布 hot strip rolling rolling characteristic analysis finite element method thickness distribution 在热连轧粗轧生产过程中，中间坯厚度是重要的产品质量控制指标，控制好中间坯厚度是实现成品轧件厚度高精度控制的前提，能够为精轧区穿带过程的稳定性提供基础条件。但在实际生产过程中，中间坯厚度一般超过35 mm，X射线测厚仪难以进行准确测量，在粗轧机出口侧一般不安装测厚仪进行在线厚度测量 [ 1 1 - 2 2 1-2 ] ；平辊轧制一般采用恒定辊缝进行厚度控制，加热炉内板坯受热不均匀以及轧制过程中头尾的温度差异，会对最终成品全长的厚度精度产生直接影响 [ 3 3 - 4 4 3-4 ] 。 目前，有关板带热轧粗轧轧制过程的研究工作多侧重于板带的横向金属流动研究，针对长度方向上的厚度分布的研究较少。苗雨川 [ 5 5 ] 提出了对板坯水印位置的轧制制度进行单独设定的方法，有效减小了加热炉水印对板坯厚度精度的影响。王四海等 [ 6 6 ] 基于粒子群差分进化算法对粗轧负荷分配进行优化，提高了中间坯厚度的精度，减小了中间坯的横向厚度差。王学谦 [ 7 7 ] 建立了厚度控制模型，提出了优化轧辊补偿曲率曲线，提高了中间坯头尾厚度的控制精度。为获得粗轧过程中的温度分布，易成新等 [ 8 8 ] 提出一种基于随机森林-长短期神经网络的板坯粗轧出口温度预测模型，提高了热连轧过程中数据输入的时间序列特性和板坯温度预测的精度与实时性。SHAO等 [ 9 9 ] 根据能量守恒和有限差分法，建立了一种轧件瞬态温度模型，分析了不同轧件厚度、宽度和初始横向温差对轧件横向温度的影响规律，发现轧件出口的横向温度分布与厚度、宽度和轧件初始横向温差成正比。JANG等 [ 10 10 ] 分析了粗轧后轧件在保温罩内的温度变化，研究了在保温罩内的轧件头尾温差变化规律。随着计算机技术的迅速发展，有限元仿真在轧制过程板带金属流动规律的研究中得到广泛应用。曹建国等 [ 11 11 ] 建立了三维六辊轧机辊系弹性变形有限元模型，分析了UCM轧机在冷轧过程中对带钢板形控制的影响。SONG等 [ 12 12 ] 应用ABAQUS有限元软件建立了平整过程带钢轧辊有限元模型，研究了不同磨损阶段工作辊与带钢间接触压力的分布规律。张清东等 [ 13 13 ] 通过ANSYS有限元软件求解了六辊CVC轧机的三维变形，讨论了轧辊变形对板带凸度的影响。CHAI等 [ 14 14 ] 用有限元方法计算了多种情况下轧件的金属流动规律，对热轧各机架凸度进行了优化。YAO等 [ 15 15 ] 建立了一种实时的热轧带钢三维金属流动模型，并通过ABQUS有限元软件对模型进行了验证。WANG等 [ 16 16 ] 通过有限元模拟，研究了带钢金属横向流对板形分布的影响规律。 为准确得到热连轧中间坯长度方向上的厚度分布，本文通过轧制特性对中间坯长度方向厚度偏差产生的原因进行系统分析，通过影响系数法计算得到入口温度偏差和厚度偏差对出口厚度偏差的影响规律；进一步使用ABAQUS建立三维有限元热力耦合模型，对粗轧过程中金属流动规律进行仿真和验证，得到中间坯长度方向的厚度和温度分布情况，以期为后续精轧区厚度控制提供准确的源数据，确保生产过程的稳定进行。 1 厚度偏差产生的原因 1 在板带热轧生产过程中，板坯经加热炉加热之后，沿运输辊道经高压水除鳞后到达粗轧机组，通过奇数道次的可逆轧制得到满足要求的中间坯后，进入精轧机组完成轧制过程。典型的热连轧工艺流程如 图1 图1 所示。温度是影响轧制的重要因素之一，对轧制变形产生直接的影响 [ 17 17 - 19 19 17-19 ] 。在加热炉加热过程中，板坯与步进梁接触位置的温度一般低于未与步进梁接触位置的温度 [ 20 20 ] ；而且坯料在加热炉内加热时，坯料表面会形成分布不均匀的结垢层和氧化铁皮，不仅会影响坯料加热均匀性，同时会导致板坯出炉温度难以在线准确测量；经过高压水除鳞后，板坯上的氧化铁皮会被清除，但由于板坯表面会出现返红现象，导致板坯温度仍难以被准确检测 [ 21 21 ] 。 随着热轧过程的进行，轧件的温度变化主要包括与空气发生热辐射、对流造成空冷温降、高压水除鳞过程中的水冷温降、轧件变形过程中与轧辊的摩擦换热、接触换热以及轧件的塑性变形热 [ 22 22 ] 。由于轧件头部先于轧件尾部发生变形，轧件尾部发生变形时的温度比轧件头部的低，因此，沿轧制方向会产生趋势性的温度偏差。在轧制过程中，轧件温度变化示意图如 图2 图2 所示。 轧件空冷温降模型为 <math display="block" id="M1"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">a</mi></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><mi>T</mi><mo>+</mo><mn mathvariant="normal">237</mn><mo stretchy="false">)</mo><mo>-</mo><mn mathvariant="normal">100</mn><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">6</mn><mi>ε</mi><mi>δ</mi></mrow><mrow><mn mathvariant="normal">100</mn><mi>γ</mi><mi>c</mi><mi>h</mi></mrow></mfrac><mi>τ</mi><mo>+</mo><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mi>T</mi><mo>+</mo><mn mathvariant="normal">273</mn></mrow><mrow><mn mathvariant="normal">100</mn></mrow></mfrac></mrow></mfenced></mrow><mrow><mo>-</mo><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfenced></mrow><mrow><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></mfrac></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679548&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679536&type= 72.38999939 12.78466606 式中： <math id="M2"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">a</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679552&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679550&type= 5.50333309 3.72533321 为空冷温降，℃； <math id="M3"><mi>T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679554&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679537&type= 2.45533323 2.62466669 为轧件温度，℃； <math id="M4"><mi>ε</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679541&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679539&type= 1.52400005 2.62466669 为辐射率； <math id="M5"><mi>δ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679558&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679556&type= 1.77800000 2.62466669 为玻尔兹曼常数； <math id="M6"><mi>τ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679560&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679544&type= 1.43933344 2.62466669 为时间，s； <math id="M7"><mi>γ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679564&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679562&type= 1.52400005 3.38666677 为轧件密度，kg/m 3 ； <math id="M8"><mi>h</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679547&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679566&type= 1.86266661 2.62466669 为轧件厚度，mm； c 为比热容，J/(kg·℃)。 轧件水冷温降模型为 <math display="block" id="M9"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">w</mi></mrow></msub><mo>=</mo><mfrac><mrow><mo>-</mo><mn mathvariant="normal">2</mn><mi>α</mi></mrow><mrow><mi>γ</mi><mi>c</mi></mrow></mfrac><mo>⋅</mo><mi>τ</mi><mfenced separators="|"><mrow><mi>T</mi><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">w</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679568&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679567&type= 33.86666870 8.04333401 式中： <math id="M10"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">w</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679582&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679569&type= 6.09600019 3.72533321 为水冷温降，℃； <math id="M11"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679572&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679583&type= 1.94733346 2.62466669 为换热系数， W/(m 2 ·℃)； <math id="M12"><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">w</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679576&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679574&type= 3.89466691 3.72533321 为冷却水温度，℃。 轧件与轧辊间的摩擦温升模型为 <math display="block" id="M13"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">f</mi></mrow></msub><mo>=</mo><mi>A</mi><mo>⋅</mo><mfrac><mrow><mi>f</mi></mrow><mrow><mi>γ</mi><mi>c</mi></mrow></mfrac><mo>⋅</mo><mfrac><mrow><msubsup><mrow><mi>l</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow><mrow><mi>'</mi></mrow></msubsup></mrow><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub></mrow></mfrac><mi>K</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mfrac><mrow><mi>H</mi></mrow><mrow><mi>h</mi></mrow></mfrac><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679594&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679593&type= 45.55066681 8.21266651 式中： <math id="M14"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">f</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679597&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679586&type= 5.24933338 3.72533321 为摩擦温升，℃； A 为系数； <math id="M15"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679588&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679598&type= 2.37066650 3.38666677 为摩擦因数； <math id="M16"><msub><mrow><mi>h</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679590&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679589&type= 3.80999994 3.72533321 为平均厚度，mm； <math id="M17"><msubsup><mrow><mi>l</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow><mrow><mi>'</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679604&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679602&type= 2.20133328 4.06400013 为变形区接触弧长度，mm； <math id="M18"><mi>K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679607&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679605&type= 2.79399991 2.62466669 为变形抗力，MPa； <math id="M19"><mi>H</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679610&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679623&type= 3.13266683 2.62466669 为入口厚度，mm。 轧件的塑性变形温升模型为 <math display="block" id="M20"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">h</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi>p</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mfrac><mrow><mi>H</mi></mrow><mrow><mi>h</mi></mrow></mfrac><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">427</mn><mi>γ</mi><mi>c</mi></mrow></mfrac><mi>η</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679626&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679625&type= 31.24200058 11.85333347 式中： <math id="M21"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">h</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679629&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679628&type= 5.67266655 3.72533321 为塑性变形温升，℃； p 为平均单位压力，kN； <math id="M22"><mi>η</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679612&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679631&type= 1.86266661 3.38666677 为轧件温升在变形热中的占比； 轧件与轧辊接触温降模型为 <math display="block" id="M23"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">4</mn><mi>β</mi><mfrac><mrow><msubsup><mrow><mi>l</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow><mrow><mi>'</mi></mrow></msubsup></mrow><mrow><mi>h</mi></mrow></mfrac><mroot><mrow><mfrac><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">π</mi><msubsup><mrow><mi>l</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow><mrow><mi>'</mi></mrow></msubsup><mi>v</mi></mrow></mfrac></mrow><mrow/></mroot><mo>⋅</mo><mfenced separators="|"><mrow><mi>T</mi><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">R</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679616&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679614&type= 42.84133148 9.82133293 式中： <math id="M24"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679634&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679618&type= 5.50333309 3.72533321 为轧件与轧辊接触温降，℃； <math id="M25"><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">R</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679637&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679635&type= 3.72533321 3.72533321 为轧辊温度，℃； <math id="M26"><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679639&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679621&type= 2.11666679 3.38666677 为热传导效率； <math id="M27"><mi>v</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679654&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679640&type= 1.69333339 2.62466669 为轧制速度，m/s。 由于轧件温度分布的差异，变形过程中变形抗力发生变化，在同样的工艺条件下，轧件出口厚度也会产偏差，所产生的厚度偏差会随着轧制过程的进行而传到后续道次或工序，从而影响最终产品的厚度分布 [ 23 23 - 25 25 23-25 ] 。即使在初始坯料没有厚度偏差的情况下，加热炉加热过程和后续的轧制过程中产生的温度偏差也会导致轧件产生厚度偏差，最终影响中间坯全长方向上的厚度分布。 2 轧制特性分析 1 2.1 　厚度增量模型 2 在轧制过程中，将弹跳增量方程代入轧制力模型 <math id="M28"><msub><mrow><mi>P</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679658&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679656&type= 5.75733376 3.72533321 f ( <math id="M29"><msub><mrow><mi>h</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679662&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679660&type= 2.70933342 3.72533321 ， <math id="M30"><msub><mrow><mi>H</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679665&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679664&type= 3.55599999 3.72533321 ， <math id="M31"><msub><mrow><mi>K</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679644&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679642&type= 3.38666677 3.72533321 ， <math id="M32"><msub><mrow><mi>τ</mi></mrow><mrow><mi mathvariant="normal">F</mi><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679647&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679645&type= 3.38666677 3.72533321 ， <math id="M33"><msub><mrow><mi>τ</mi></mrow><mrow><mi mathvariant="normal">b</mi><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679670&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679649&type= 3.30200005 3.72533321 )，获得增量的厚度方程。由于粗轧过程采用恒定辊缝，并且为无张力轧制，因此，粗轧过程厚度增量方程可简化为 <math display="block" id="M34"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>h</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>H</mi><mo>,</mo><mi>i</mi></mrow></msub><mi mathvariant="normal">Δ</mi><msub><mrow><mi>H</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>K</mi><mo>,</mo><mi>i</mi></mrow></msub><mi mathvariant="normal">Δ</mi><msub><mrow><mi>K</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679683&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679651&type= 34.12067032 3.72533321 式中： <math id="M35"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>h</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679687&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679685&type= 4.91066647 3.72533321 为第 i 道次出口厚度增量，mm； <math id="M36"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>H</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679689&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679673&type= 5.75733376 3.72533321 为第 i 道次入口厚度增量，mm； <math id="M37"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>K</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679676&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679675&type= 5.58799982 3.72533321 为第 i 道次变形抗力增量，MPa； <math id="M38"><msub><mrow><mi>A</mi></mrow><mrow><mi>H</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mfenced separators="|"><mrow><mfrac><mrow><mrow><mrow><mo>∂</mo><mi>P</mi></mrow><mo>/</mo><mrow><mo>∂</mo><mi>H</mi></mrow></mrow></mrow><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">P</mi></mrow></msub><mo>-</mo><mrow><mrow><mo>∂</mo><mi>P</mi></mrow><mo>/</mo><mrow><mo>∂</mo><mi>h</mi></mrow></mrow></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679693&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679679&type= 30.47999954 9.90600014 ，为第 i 道次入口厚度影响系数； <math id="M39"><msub><mrow><mi>A</mi></mrow><mrow><mi>K</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mfenced separators="|"><mrow><mfrac><mrow><mrow><mrow><mo>∂</mo><mi>P</mi></mrow><mo>/</mo><mrow><mo>∂</mo><mi>K</mi></mrow></mrow></mrow><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">P</mi></mrow></msub><mo>-</mo><mrow><mrow><mo>∂</mo><mi>P</mi></mrow><mo>/</mo><mrow><mo>∂</mo><mi>h</mi></mrow></mrow></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679680&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679695&type= 30.39533424 9.90600014 ，为第 i 道次变形抗力影响系数； <math id="M40"><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">P</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679713&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679698&type= 3.89466691 3.72533321 为轧机刚度系数，kN/mm； H 为入口厚度； h 为出口厚度； P 为轧制压力，kN。 各个偏微分系数可以用以弦代切线的方法，在小范围内将非线性函数以线性化的方式进行计算，如 图3 图3 所示。 图3 图3 中， h R 为轧辊辊缝宽度， h 0 为入口厚度；设 a 点的轧制力为 P ，厚度为 h 1 ， b 点的轧制力为 <math id="M41"><mi>P</mi><mi mathvariant="normal">'</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679716&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679699&type= 3.21733332 2.62466669 ，厚度为 h 2 ，在小范围内可以将 a 点和 b 点的非线性函数线性化，则 <math id="M42"><mfrac><mrow><mo>∂</mo><mi>P</mi></mrow><mrow><mo>∂</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>a</mi></mrow></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mrow><mi>P</mi></mrow><mrow><mi>'</mi></mrow></msup><mo>-</mo><mi>P</mi></mrow><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>-</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>a</mi></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679720&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679705&type= 20.82799911 8.12800026 ，用同样的方法可以计算出其他一系列相应的偏微分系数。 2.2 　影响系数分析 2 为了分析入口温度偏差和入口厚度偏差对成品厚度的影响，建立由入口温度、入口厚度和成品厚度组成的静态综合分析增量数学模型，并求出相应的影响系数 [ 26 26 - 27 27 26-27 ] 。 2.2.1　温度偏差对出口厚度的影响系数 3 轧件长度方向上的初始温度偏差不仅会影响第1道次的出口厚度，还会对后续各道次的出口厚度造成影响。当研究初始温度偏差对出口厚度的影响时，温度的变化导致变形抗力的变化，从而对出口厚度产生影响。温度偏差对厚度的影响系数 <math id="M43"><msubsup><mrow><mi>K</mi></mrow><mrow><msub><mrow><mi>K</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679730&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679729&type= 5.16466665 4.82600021 可由下式计算： <math display="block" id="M44"><msubsup><mrow><mi>K</mi></mrow><mrow><msub><mrow><mi>K</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msubsup><mo>=</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>A</mi></mrow><mrow><mi>k</mi><mo>,</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>⋅</mo><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>H</mi><mo>,</mo><mi>i</mi></mrow></msub></mrow><mo>+</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679737&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679734&type= 33.61266708 8.72066593 <math display="block" id="M45"><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>A</mi></mrow><mrow><mi>k</mi><mo>,</mo><mn mathvariant="normal">2</mn></mrow></msub><mo>⋅</mo><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">2</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>H</mi><mo>,</mo><mi>i</mi></mrow></msub></mrow><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi></mrow></msub><msub><mrow><mi>A</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>n</mi></mrow></msub><mo>⋅</mo><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mi>n</mi></mrow><mrow/></munderover></mstyle><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>H</mi><mo>,</mo><mi>i</mi></mrow></msub></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679742&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679739&type= 54.77933884 8.72066593 式中： <math id="M46"><msubsup><mrow><mi>K</mi></mrow><mrow><msub><mrow><mi>K</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679762&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679751&type= 5.16466665 4.82600021 为温度偏差对成品厚度的影响系数； <math id="M47"><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679765&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679755&type= 2.53999996 3.72533321 为初始温度偏差对后续道次变形抗力的影响系数。 2.2.2　厚度偏差对出口厚度的影响系数 3 由于轧件在长度方向上出现初始温度偏差，当经过第1道次轧制之后，轧件的出口厚度在长度方向上开始出现偏差，厚度偏差又会对后续道次产生影响，因此，还需要对各道次入口厚度偏差对成品厚度的影响进行分析。入口厚度对各道次出口厚度的影响系数为 <math display="block" id="M48"><msubsup><mrow><mi>K</mi></mrow><mrow><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msubsup><mo>=</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>H</mi><mo>,</mo><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>A</mi></mrow><mrow><mi>H</mi><mo>,</mo><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>A</mi></mrow><mrow><mi>H</mi><mo>,</mo><mn mathvariant="normal">3</mn></mrow></msub><mo>⋯</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>H</mi><mo>,</mo><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679778&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679773&type= 36.91466904 5.92666674 式中： <math id="M49"><msubsup><mrow><mi>K</mi></mrow><mrow><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679860&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679864&type= 5.24933338 4.82600021 为入口厚度偏差对成品厚度的影响系数。 2.2.3　影响系数分析与计算 3 以 图1 图1 所示热连轧粗轧实际生产过程为例，轧制总道次为5道次，工作辊直径为900 mm，轧件宽度为1 250 mm，轧机刚度为5 000 kN/mm，钢种为Q235，中间坯厚度为38 mm。当初始入口厚度偏差Δ H 0 =1 mm，温度偏差Δ T 0 =20 ℃时，对粗轧过程进行轧制特性分析，计算得到的偏微分系数如 表1 表1 所示。 2 参数 5 道次 R1 R2 R3 R4 R5 入口厚度 H /mm 230 180 132 87 57 出口厚度 h /mm 180 132 87 57 38 轧制速度 v /(m·s -1 ) 2.0 3.0 3.3 3.6 4.0 轧制力 P /kN 15 283 16 987 19 683 20 367 20 151 变形抗力 K /MPa 71.03 87.68 103.46 120.47 137.50 <math id="M50"><mrow><mrow><mo>∂</mo><mi>P</mi></mrow><mo>/</mo><mrow><mo>∂</mo><mi>H</mi></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679790&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679772&type= 8.55133343 2.79399991 /(kN·mm -1 ) 173.4 191.4 302.3 472.6 759.4 <math id="M51"><mrow><mrow><mo>∂</mo><mi>P</mi></mrow><mo>/</mo><mrow><mo>∂</mo><mi>h</mi></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679795&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679793&type= 7.87400007 2.79399991 /(kN·mm -1 ) -168.8 -185.2 -398.2 -640.7 -1 035.3 <math id="M52"><mrow><mrow><mo>∂</mo><mi>P</mi></mrow><mo>/</mo><mrow><mo>∂</mo><mi>K</mi></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679799&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679807&type= 8.38199997 2.79399991 /(kN·MPa -1 ) 212.2 187.8 186.2 169.8 145.9 中间坯厚度为38 mm的轧件影响系数 <math id="M53"><msubsup><mrow><mi>K</mi></mrow><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679814&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679821&type= 4.82600021 4.82600021 和 <math id="M54"><msubsup><mrow><mi>K</mi></mrow><mrow><msub><mrow><mi>K</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679816&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679825&type= 5.16466665 4.82600021 的计算结果分别如 图4 图4 和 图5 图5 所示。从 图4 图4 可以看出：随着粗轧的进行，各道次入口温度偏差对出口厚度的影响系数越来越小。在第1道次，出口厚度影响系数为0.041；在第5道次，出口厚度影响系数减小为0.018。从 图5 图5 可以看出：随着粗轧过程的进行，轧件的初始温度偏差对各道次出口厚度的影响系数越来越小；初始温度偏差对第1道次的出口厚度影响系数为0.041，对第5道次的出口厚度影响系数为0.022。 中间坯厚度为38 mm的轧件影响系数 <math id="M55"><msubsup><mrow><mi>K</mi></mrow><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679859&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679857&type= 4.91066647 4.82600021 和 <math id="M56"><msubsup><mrow><mi>K</mi></mrow><mrow><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679860&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679864&type= 5.24933338 4.82600021 的计算结果分别如 图6 图6 和 图7 图7 所示。从 图6 图6 可以看出：随着粗轧过程的进行，入口厚度的偏差对出口厚度的影响系数越来越大；在第1道次，入口厚度偏差对出口厚度的影响系数为0.034；在第5道次，入口厚度偏差对出口厚度的影响系数为0.124，对中间坯厚度的影响较大。从 图7 图7 可以看出：随着粗轧过程的进行，轧件的初始厚度偏差对各道次出口厚度的影响系数越来越小；第1道次初始厚度偏差对出口厚度的影响系数为0.034，对第5道次的出口厚度的影响系数已经基本为0。 2.3 　厚度分布计算 2 由2.2节分析可知轧件各道次的入口温度偏差和入口厚度偏差都会对各道次出口厚度造成影响，通过温降方程计算得到轧件长度方向上不同位置的温度分布，结合温度偏差和入口厚度偏差对出口厚度的影响系数，可以计算轧件各道次由入口温度偏差和入口厚度偏差造成的总出口厚度偏差。 对于轧件长度方向温度偏差引起的厚度偏差可以由下式计算： <math display="block" id="M57"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>h</mi></mrow><mrow><mi>K</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>=</mo><msubsup><mrow><mi>K</mi></mrow><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msubsup><mo>⋅</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679901&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679899&type= 22.43666649 5.92666674 式中： <math id="M58"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>h</mi></mrow><mrow><mi>K</mi><mo>,</mo><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679909&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679903&type= 7.19666624 3.72533321 为第 i 道次由于温度偏差造成的各道次轧件长度方向上各位置的厚度偏差，mm； <math id="M59"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679923&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679912&type= 5.16466665 3.72533321 为粗轧第 i 道次轧件长度方向各位置的温度偏差，℃。 当轧件通过粗轧第1道次以后，因为轧件长度方向上的初始温度偏差造成的厚度偏差在第1道次出口就会出现，并且这种厚度偏差还会继续影响后续各道次的出口厚度，最终体现在中间坯长度方向上厚度的分布不均匀。第2道次后由轧件入口厚度偏差引起的各道次轧件长度方向上各位置的厚度偏差可由下式计算： <math display="block" id="M60"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>h</mi></mrow><mrow><mi>H</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>=</mo><msubsup><mrow><mi>K</mi></mrow><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msubsup><mi mathvariant="normal">Δ</mi><msub><mrow><mi>H</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679925&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679915&type= 21.42066574 5.92666674 式中： <math id="M61"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>h</mi></mrow><mrow><mi>H</mi><mo>,</mo><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679932&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679927&type= 7.28133297 3.72533321 为粗轧第 i 道次轧件长度方向上入口厚度偏差造成的各道次长度方向上的厚度偏差，mm； <math id="M62"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>H</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679943&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679934&type= 5.75733376 3.72533321 为各粗轧第 i 道次轧件长度方向上入口厚度的偏差，mm。 联合 式(4) 式(4) 和 式(5) 式(5) 可以求出中间坯长度方向上各位置总的厚度偏差，进而得到中间坯厚度在长度方向上的分布情况。 <math display="block" id="M63"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>h</mi></mrow><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">l</mi></mrow></msub><mo>=</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>h</mi></mrow><mrow><mi>H</mi><mo>,</mo><mn mathvariant="normal">5</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>h</mi></mrow><mrow><mi>K</mi><mo>,</mo><mn mathvariant="normal">5</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679958&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679945&type= 28.70199966 3.72533321 式中： <math id="M64"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>h</mi></mrow><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">l</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679952&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679960&type= 6.68866634 3.72533321 为中间坯长度方向各位置厚度总偏差，mm。 中间坯厚度为中间坯目标与中间坯长度方向各位置厚度总偏差之和： <math display="block" id="M65"><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo>=</mo><msub><mrow><mi>h</mi></mrow><mrow><mi mathvariant="normal">a</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>h</mi></mrow><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">l</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679983&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679970&type= 21.75933456 3.72533321 式中： <math id="M66"><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679987&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679975&type= 3.21733332 3.72533321 为中间坯实际厚度，mm； <math id="M67"><msub><mrow><mi>h</mi></mrow><mrow><mi mathvariant="normal">a</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679994&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679990&type= 5.41866684 3.72533321 为中间坯目标厚度，mm。 3 有限元仿真 1 采用有限元软件ABAQUS/Explicit建立粗轧过程动态热力耦合有限元模型，分析温度偏差和厚度偏差对成品厚度的影响。 3.1 　材料模型和仿真参数 2 为验证轧制特性分析方法计算的准确性，建立粗轧过程有限元仿真模型，将仿真模型所得结果与计算结果进行对比。为了更加符合现场实际情况，建立三维动态热力耦合有限元模型。由于轧辊轧件上下对称，仅建立轧辊带钢1/2模型，在轧辊与轧件接触区域进行局部的网格细化，以提高计算效率，见 图8 图8 。在轧制过程中，仿真建模的几何参数和轧制过程工艺参数与 表1 表1 中的一致，轧件材料属性采用Q235，轧件的材料性能和热物性参数采用JMatPro软件进行计算。带钢材料性能和热物性参数如 图9 图9 所示。 (a) 弹性模量和泊松比；(b) 导热系数；(c) 真应力-真应变；(d) 比热容 3.2 　边界条件和施加载荷 2 通过在轧辊辊径两端建立刚性体，通过刚性端部带动轧辊匀速旋转，约束刚性端部所有的位移自由度和除轧辊轴心外的所有旋转自由度，在轧件的厚度方向上施加对称约束。给定轧辊初始辊缝，轧辊以恒定速度旋转，轧件以略小于工作辊线速度的初速度向轧辊移动，使带钢能够顺利咬入。为了模拟轧制过程中轧件温度的变化，设置轧制过程中轧件的空冷换热系数为0.18 kW/(m 2 ·℃)，轧件的表面辐射率为0.8，轧辊与轧件之间的热交换系数30 kW/(m 2 ·℃)。在轧制过程中，轧件的塑性变形功大部分被转换成热量，转换系数设为0.9。由于板坯的出炉温度不易测量，结合文献[ 28 28 ]，轧件长度方向上出炉温度分布如 图10 图10 所示。 图10 图10 中，长度百分比表征在轧件长度方向上的取样位置(0代表轧件头部，100%代表轧件尾部)。假设轧件宽度方向上的温度分布一致，温度取平均值，在轧制过程中，轧制工况计算过程均与文献[ 28 28 ]中的计算过程保持一致。 3.3 　有限元模型验证 2 为验证有限元模型的准确性，将生产过程中的轧制力计算值与有限元模拟值进行对比，结果如 表2 表2 所示。由 表2 表2 可以看出，第1道次至第5道次的轧制力计算值与模拟值相对误差分别为-3.51%、2.25%、3.31%、-1.97%和4.21%，最大相对误差仅为4.21%，由此验证了本文所建立的有限元模型的准确性。 道次 轧制力有限元模拟值/kN 轧制力计算值/kN 相对误差/% R1 15 820 15 283 -3.51 R2 16 605 16 987 2.25 R3 19 031 19 683 3.31 R4 20 769 20 367 -1.97 R5 19 303 20 151 4.21 4 分析与讨论 1 4.1 　中间坯长度方向上的温度分布规律 2 轧制过程中奇数道次出口温度分布有限元仿真结果和计算结果对比如 图11 图11 所示。从 图11 图11 可见：随着轧制过程的进行，有限元仿真结果和计算结果呈相同的变化趋势，各道次出口温度逐渐降低，轧件头部和尾部的温差越来越大，沿轧件长度方向的温度有下降的趋势，轧件在有步进梁位置和无步进梁位置的温差逐渐减小；在R1道次时，轧件头部和尾部的基本没有温差；在R5道次结束时，轧件头部和尾部的出口温差为21.73 ℃；而在有步进梁和无步进梁位置的出口温差在R5道次结束时为32.14 ℃。 4.2 　入口温度偏差和入口厚度偏差对出口厚度的影响 2 为了进一步了解入口温度偏差和入口厚度偏差对出口厚度的影响，对粗轧过程R5道次厚度偏差与长度百分比的关系进行分析，结果如 图12 图12 所示。由 图12 图12 可见：由入口温度偏差造成的厚度偏差最大为0.484 mm，由于入口厚度偏差导致的出口厚度偏差最大为0.044 mm，由入口温度偏差造成的出口厚度偏差占总厚度偏差的91.7%，由入口厚度偏差造成的出口厚度偏差占总厚度偏差的8.3%。由此可见，温度偏差对中间坯厚度偏差的影响起主导作用。 4.3 　中间坯长度方向上厚度分布规律 2 中间坯厚度在长度方向上的分布规律如 图13 图13 所示。从 图13 图13 可以看出：中间坯长度方向即从头部至尾部厚度偏差呈增大趋势，带钢头部和尾部的厚度偏差为0.20 mm；与步进梁接触的位置温度较低，此处中间坯的厚度偏差较大，最末一个步进梁造成的温度低点对应的中间坯厚度偏差达到 0.53 mm，中间坯长度方向上的厚度与目标厚度的相对误差最大为1.4%左右。 5 结论 1 1) 系统分析了板带热轧生产过程中中间坯厚度产生偏差的原因。采用轧制特性分析的方法研究了温度偏差和厚度偏差对中间坯厚度的影响规律。温度偏差对厚度偏差的影响具有遗传性，厚度偏差随着轧制过程的进行对厚度偏差的影响逐渐减小。 2) 温度偏差对中间坯厚度的影响大于厚度偏差造成的影响。随着轧制的进行，中间坯厚度呈变大趋势，中间坯头尾的厚度偏差为0.20 mm，厚度偏差最大达到0.53 mm，中间坯长度方向上的厚度与目标厚度的最大相对误差可达1.4%左右。 参考文献 常蕾 , 党红文 , 麻硕 , 等 . 一种X射线测厚仪在线校正的方法 [J]. 自动化与仪器仪表 , 2015 ( 8 ): 32 - 34 . 一种X射线测厚仪在线校正的方法 CHANG Lei , DANG Hongwen , MA Shuo , et al . 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