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Effect of strain rate difference between inside and outside groove in M-K model on prediction of forming limit curve of Ti6Al4V at elevated temperatures 被引量:4
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作者 Xiao-qiang LI Hong-rui DONG +2 位作者 Hai-bo WANG Gui-qiang GUO Dong-sheng LI 《Transactions of Nonferrous Metals Society of China》 SCIE EI CAS CSCD 2020年第2期405-416,共12页
The influence of initial groove angle on strain rate inside and outside groove of Ti6Al4V alloy was investigated.Based on the evolution of strain rate inside and outside groove,the effect of strain rate difference on ... The influence of initial groove angle on strain rate inside and outside groove of Ti6Al4V alloy was investigated.Based on the evolution of strain rate inside and outside groove,the effect of strain rate difference on the evolution of normal stress and effective stress inside and outside groove was also analyzed.The results show that when linear loading path changes from uniaxial tension to equi-biaxial tension,the initial groove angle plays a weaker role in the evolution of strain rate in the M-K model.Due to the constraint of force equilibrium between inside and outside groove,the strain rate difference makes the normal stress inside groove firstly decrease and then increase during calculation,which makes the prediction algorithm of forming limit convergent at elevated temperature.The decrease of normal stress inside groove is mainly caused by high temperature softening effect and the rotation of groove,while the increase of normal stress inside groove is mainly due to strain rate hardening effect. 展开更多
关键词 Ti6Al4V alloy strain rate difference forming limit M-K model stress evolution
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Simulation of bulk metal forming processes using one-step finite element approach based on deformation theory of plasticity 被引量:2
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作者 王鹏 董湘怀 傅立军 《Transactions of Nonferrous Metals Society of China》 SCIE EI CAS CSCD 2010年第2期276-282,共7页
The bulk metal forming processes were simulated by using a one-step finite element(FE)approach based on deformation theory of plasticity,which enables rapid prediction of final workpiece configurations and stress/stra... The bulk metal forming processes were simulated by using a one-step finite element(FE)approach based on deformation theory of plasticity,which enables rapid prediction of final workpiece configurations and stress/strain distributions.This approach was implemented to minimize the approximated plastic potential energy derived from the total plastic work and the equivalent external work in static equilibrium,for incompressibly rigid-plastic materials,by FE calculation based on the extremum work principle.The one-step forward simulations of compression and rolling processes were presented as examples,and the results were compared with those obtained by classical incremental FE simulation to verify the feasibility and validity of the proposed method. 展开更多
关键词 bulk metal forming plastic deformation theory finite element method one-step forward simulation rigid-plastic materials
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Effects of Heating Rate on the Process Parameters of Superplastic Forming for Zr_(55)Cu_(30)Al_(10)Ni_5 被引量:1
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作者 杨璠 SHI Tielin 廖广兰 《Journal of Wuhan University of Technology(Materials Science)》 SCIE EI CAS 2014年第6期1136-1140,共5页
We investigated the effects of heating rate on the process parameters of superplastic forming for Zr55Cu30Al10Ni5 by differential scanning calorimetry. The continuous heating and isothermal annealing analyses suggeste... We investigated the effects of heating rate on the process parameters of superplastic forming for Zr55Cu30Al10Ni5 by differential scanning calorimetry. The continuous heating and isothermal annealing analyses suggested that the temperatures of glass transition and onset crystallization are heating rate-dependent in the supercooled liquid region. Then, the time-temperature-transformation diagram under different heating rates indicates that increasing the heating rate can lead to an increase of the incubation time at the same anneal temperature in the supercooled liquid region. Based on the Arrhenius relationship, we discovered that the incubation time increases by 1.08-1.11 times with double increase of the heating rate at the same anneal temperature, and then verified it by the data of literatures and the experimental results. The obtained curve of the max available incubation time reveals that the incubation time at a certain anneal temperature in the supercooled liquid region is not infinite, and will increase with increasing heating rate until this temperature shifts out of the supercooled liquid region because of exceeding critical heating rate. It is concluded that heating rate must be an important processing parameter of superplastic forming for Zr55Cu30Al10Ni5. 展开更多
关键词 bulk metallic glass annealing superplastic forming heating rate incubation time
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A method for calculation of ion distribution in reaction system forming hydroxide
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作者 金哲男 郑大录 +3 位作者 洪正秀 郑英勋 路殿坤 陈国宝 《Transactions of Nonferrous Metals Society of China》 SCIE EI CAS CSCD 2013年第12期3793-3798,共6页
A formula was proposed to calculate the distribution of metal ions quantitatively in chemical reaction system forming hydroxide where precipitation and complex are formed together. The effects of some factors on forma... A formula was proposed to calculate the distribution of metal ions quantitatively in chemical reaction system forming hydroxide where precipitation and complex are formed together. The effects of some factors on formation of precipitation and complex were investigated, and the corresponding precipitation rates of zinc, iron (III), aluminum, copper and magnesium were calculated. As a result, it shows that the proposed formula is reliable. By the proposed formula, the existence state of metal ions in hydroxides reaction system with any metal ions can be well described and the effects of some factors on the distribution of metal ions were determined. 展开更多
关键词 HYDROXIDE metal ions distribution precipitation rate complex forming rate
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Effects of N Forms and Rates on Vegetable Growth and Nitrate Accumulation 被引量:55
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作者 WANG Zhao-Hui and LI Sheng-XiuCollege of Resources and Environmental Sciences, Northwestern Science and Technology University of Agriculture and Forestry, Yangling 712100 (China). E-mail: w-zhaohui@263.net 《Pedosphere》 SCIE CAS CSCD 2003年第4期309-316,共8页
Experiments were carried out on a vegetable field with Peking cabbage(Brassica pekinensis (Lour.) Rupr.), cabbage (Brassica, chinensis var. oleifcra Makino and nemoto),green cabbage (Brassica chinensis L.), spinach (S... Experiments were carried out on a vegetable field with Peking cabbage(Brassica pekinensis (Lour.) Rupr.), cabbage (Brassica, chinensis var. oleifcra Makino and nemoto),green cabbage (Brassica chinensis L.), spinach (Spinacia oleracea L.) and rape(Brassica campestrisL.) to study the effects of N forms and N rates on their growth and nitrate accumulation. Theresults indicated that application of ammonium chloride, ammonium nitrate, sodium nitrate and ureasignificantly increased the yields and nitrate concentrations of Peking cabbage and spinach.Although no significant difference was found in the yields after application of the 4 N forms,nitrate N increased nitrate accumulation in vegetables much more than ammonium N. The vegetableyields were not increased continuously with N rate increase, and oversupply of N reduced the plantgrowth, leading to a yield decline. This trend was also true for nitrate concentrations in somevegetables and at some sampling times. However, as a whole, the nitrate concentrations in vegetableswere positively correlated with N rates. Thus, addition of N fertilizer to soil was the major causefor increases in nitrate concentrations in vegetables. Nitrate concentrations were much higher inroots, stems and petioles than in blades at any N rate. 展开更多
关键词 N form N rate nitrate accumulation vegetable yield
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A Study on Super Speed Forming of Metal Sheet by Laser Shock Waves 被引量:4
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作者 ZHOU Jian-zhong 1, YANG Ji-chang 1, ZHANG Yong-kang 1, ZHOU Ming 2 (1. School of Mechanical Engineering, Jiangsu University, Zhenjiang, 21 2013 China 2. School of Material Science and Engineering, Jiangsu University, Zhenjiang 21 2000, China) 《厦门大学学报(自然科学版)》 CAS CSCD 北大核心 2002年第S1期127-128,共2页
Metal sheet plastic deformation or forming is gener at ed through a mechanical pressure or a thermal variation. These pressure variatio ns or thermal variations can be created by a variety of means such as press form ... Metal sheet plastic deformation or forming is gener at ed through a mechanical pressure or a thermal variation. These pressure variatio ns or thermal variations can be created by a variety of means such as press form ing, hydroforming, imploding detonation and so on. According to the magnitude of the strain rates all these forming methods can be divided into quasi-static fo rming and dynamical forming. Up to now there are no reports of forming methods w ith the strain rates above 10 5sec -1, even though the exploding forming. In this article, we work on a dynamic super-speed forming method driven by lase r shock waves and advanced a novel concept of laser shock forming. The initial o bservation of the laser shock forming is done through a bugle testing with speci mens of SUS430 sheet metal, using a neodymium-glass laser of pulse energy 10J~ 3 0J and duration of 20 ns (FWHM). The investigation revealed that the plastic de formation during the laser shock forming is characterized as ultrahigh strain ra te up to 10 7sec -1. We indicate that plastic deformation increases nonlin early when the energy density of the laser varies. By investigating the hardness and residual stress of the surfaces, we conclude that laser shock forming is a combination technique of laser shock strengthening and metal forming for introdu cing a strain harden and a compressive residual stress on the surface of the wor k-piece, and the treated surface by laser shock forming has good properties in fatigue and corrosion resistance. This technique can achieve forming wit h or without mould. 展开更多
关键词 laser shock wave metal sheet forming ultrahigh strain rate residual stresses
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Effect of strain rate on microstructure and mechanical properties of spray-formed Al-Cu-Mg alloy 被引量:3
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作者 Tong SHEN Cai-he FAN +3 位作者 Ze-yi HU Qin WU Yu-meng NI Yu-zhou CHEN 《Transactions of Nonferrous Metals Society of China》 SCIE EI CAS CSCD 2022年第4期1096-1104,共9页
Transmission electron microscopy(TEM),X-ray diffraction(XRD),electron backscattered diffraction(EBSD),and tensile tests were used to study the effects of strain rates(0.1,1 and 9.1 s^(-1))on the microstructure and mec... Transmission electron microscopy(TEM),X-ray diffraction(XRD),electron backscattered diffraction(EBSD),and tensile tests were used to study the effects of strain rates(0.1,1 and 9.1 s^(-1))on the microstructure and mechanical properties of spray-formed Al-Cu-Mg alloys during large-strain rolling at 420℃.Results show that during hot rolling,the proportion of high-angle grain boundaries(HAGBs)and the degree of dynamic recrystallization(DRX)initially increase and then decrease,whereas the average grain size and dislocation density show the opposite trend with the increase of the strain rate.In addition,the number of S′phases in the matrix decreases,and the grain boundary precipitates(GBPs)become coarser and more discontinuous as the strain rate increases.When the strain rate increases from 0.1 to 9.1 s^(-1),the tensile strength of the alloy decreases from 492.45 to 427.63 MPa,whereas the elongation initially increases from 12.1%to 21.8%and then decreases to 17.7%. 展开更多
关键词 spray forming Al-Cu-Mg alloy strain rate MICROSTRUCTURE mechanical properties
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VISUAL OBSERBATION OF HCFC141b GAS HYDRATE FORMATION/DECOMPOSITION PROCESS OUTSIDE OF A TUBE 被引量:3
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作者 谢应明 郭开华 +2 位作者 樊栓狮 梁德青 顾建明 《化工学报》 EI CAS CSCD 北大核心 2003年第z1期91-97,共7页
In order to design a kind of heat exchanger suitable to the indirect-touched gas hydrate cool storage vessel, a visual observation of HCFC141b gas hydrate formation/decomposition process was presented through a self-d... In order to design a kind of heat exchanger suitable to the indirect-touched gas hydrate cool storage vessel, a visual observation of HCFC141b gas hydrate formation/decomposition process was presented through a self-designed small-scale visualization apparatus of gas hydrate cool storage. Based on the shooted photos and recorded temperatures, the formation/decomposition process of HCFC141b are described, some characteristics are concluded, and some suggestions of designing heat exchanger are indicated according to the specific characteristics of HCFC141b gas hydrate formation/decomposition process. 展开更多
关键词 into or form as it on HCFC VISUAL OBSERBATION OF HCFC141b GAS HYDrate formATION/DECOMPOSITION PROCESS OUTSIDE OF A TUBE of heat that 化工学报 from be down rate
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Numerical simulation and its application of rheological forming of titanium alloy vane disk
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作者 余敏 罗迎社 +1 位作者 彭相华 覃银辉 《中国有色金属学会会刊:英文版》 CSCD 2006年第B02期671-675,共5页
The hot Theological forming method was proposed to form the second titanium alloy vane disk. The hot rheological forming process of the TC11 titanium vane disk under a certain temperature and different strain rates wa... The hot Theological forming method was proposed to form the second titanium alloy vane disk. The hot rheological forming process of the TC11 titanium vane disk under a certain temperature and different strain rates was investigated by using the bulk forming software of DEFORM 3D. A series of results including temperature field, equivalent strain distribution, load-stroke curve and rheology procedure were obtained by this finite element method. The rheological forming characteristics were well realized and the forming parameters were determined. The results and analysis show that with decreasing strain rate, the metal flow more equably and the filling of the vane shape is also better. Moreover, the mechanical properties and microstructures of the products produced by this new technique are improved evidently compared with that produced by traditional method. 展开更多
关键词 钛合金叶片桨盘 热流变成型 数值模拟 应用 锻造
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Forming Suite在车身同步工程中的应用
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作者 张骥 李旭东 《汽车实用技术》 2021年第22期152-154,共3页
建立车身同步工程可以有效地提高产品的可制造性(DFM)、降低产品成本,达到提质增效、提高收益的目的。文章从设计可制造性、成本优化两个方面,通过实例介绍Forming Suite在车身同步工程中的应用,得出基于Forming Suite的车身同步工程不... 建立车身同步工程可以有效地提高产品的可制造性(DFM)、降低产品成本,达到提质增效、提高收益的目的。文章从设计可制造性、成本优化两个方面,通过实例介绍Forming Suite在车身同步工程中的应用,得出基于Forming Suite的车身同步工程不需要专业工艺分析团队,且具有更高同步性的特点的结论。 展开更多
关键词 同步工程 材料利用率 forming Suite
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大截面变化率铝合金异形管整体液力成形工艺设计与优化
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作者 陈明 刘鑫 +5 位作者 徐勇 解文龙 黄新越 王晟诚 赵长坚 张士宏 《精密成形工程》 北大核心 2025年第1期43-50,共8页
目的针对某大截面变化率空心构件成形难的问题,采用液力成形的方法进行成形并对其生产工艺进行优化。方法首先针对目标零件,通过压扁截面处对称及圆截面处对称2种不同的对称方式设计了A型结构和B型结构,通过有限元模拟对比了同一加载路... 目的针对某大截面变化率空心构件成形难的问题,采用液力成形的方法进行成形并对其生产工艺进行优化。方法首先针对目标零件,通过压扁截面处对称及圆截面处对称2种不同的对称方式设计了A型结构和B型结构,通过有限元模拟对比了同一加载路径下2种不同结构的成形特点,选取了成形效果较好的A型结构进行成形工艺研究,建立了A型结构的成形分区图,继而根据成形分区图进行了正交试验设计。结果在同一种加载路径下,A型结构成形的目标零件最大减薄率明显小于B型结构成形的目标零件最大减薄率,且在成形过程中,B型结构压扁特征处易出现褶皱或起皱趋势,而A型结构压扁特征处壁厚分布更加均匀,A型结构成形效果优于B型结构成形效果。优化后的工艺参数如下:压制压力为2 MPa、初始压力为7 MPa、成形压力为15 MPa、轴向进给为13 mm,成形零件最大减薄率为15.9%。结论经成形分区图指导设计的正交成形试验结果均符合设计要求,成形分区图的设立对成形试验的设计具有指导作用;采用优化后的工艺参数获得了较好的试验效果,可为工业生产提供指导。 展开更多
关键词 大截面变化率 管材液力成形 结构设计 正交试验 工艺优化
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Microstructure evolution of spray-formed hypereutectic Al-Si alloys in semisolid reheating process 被引量:4
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作者 朱学卫 王日初 +1 位作者 彭建 彭超群 《Transactions of Nonferrous Metals Society of China》 SCIE EI CAS CSCD 2014年第6期1766-1772,共7页
The Al-27%Si alloy was prepared by the spray forming process,and its microstructure evolution during the semisolid reheating process was investigated.The results show that,the primary Si phase coarsens during the rehe... The Al-27%Si alloy was prepared by the spray forming process,and its microstructure evolution during the semisolid reheating process was investigated.The results show that,the primary Si phase coarsens during the reheating process and the coarsening rate increases with the increase of reheating temperature.The eutectic phase is produced in the molten region when quenched in the cold water.The microstructure evolution in the semisolid state can be divided into three stages.The remarkable characteristic of the first stage is only a solid-state phase transformation process.However,the region around the α(Al) matrix gradually melts in the second stage.The primary Si in the liquid phase coarsens obviously,and the eutectic phase is produced in the molten region when the specimens are quenched in cold water.In the last stage,the same thing as that in the second stage happens except that all the α(Al) matrixes are melted. 展开更多
关键词 spray forming Al-Si alloy SEMISOLID coarsening rate
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基于Dynaform的汽车左后侧围外板起皱问题的分析与解决 被引量:5
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作者 熊文韬 刘泓滨 +1 位作者 孙元贵 邓利军 《模具工业》 2017年第5期24-28,共5页
以某汽车左后侧围外板拉深成形为例,针对制件拉深时出现的严重起皱问题,利用Dynaform非线性有限元软件对制件拉深成形过程进行有限元分析,预测制件起皱趋势。运用UG软件对模具结构、板料尺寸以及拉深筋布置3个方面进行优化,最终制件起... 以某汽车左后侧围外板拉深成形为例,针对制件拉深时出现的严重起皱问题,利用Dynaform非线性有限元软件对制件拉深成形过程进行有限元分析,预测制件起皱趋势。运用UG软件对模具结构、板料尺寸以及拉深筋布置3个方面进行优化,最终制件起皱得到明显改善,最大增厚率由34.3%降至2.5%,符合增厚率<5%的生产要求。 展开更多
关键词 汽车左后侧围外板 拉深成形 起皱 Dyanform 最大增厚率
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基于RoCoF的构网型变流器惯量阻尼辨识方法
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作者 张盛 方支剑 +1 位作者 熊亮雳 游力 《电力电子技术》 2025年第2期84-87,共4页
针对当前变流器惯量测试方法较复杂,难以适用于构网型变流器的问题,提出基于频率变化率(RoCoF)的构网型变流器惯量阻尼辨识方法,分析了构网型变流器有功控制的工作机理,推导了构网型变流器有功功率关于电网频率斜坡扰动时的关系式,并依... 针对当前变流器惯量测试方法较复杂,难以适用于构网型变流器的问题,提出基于频率变化率(RoCoF)的构网型变流器惯量阻尼辨识方法,分析了构网型变流器有功控制的工作机理,推导了构网型变流器有功功率关于电网频率斜坡扰动时的关系式,并依据该关系式阐述了惯量和阻尼辨识步骤。最后,在3.15 MW的构网型电化学储能机组中对所提方法进行了验证,实验结果表明所提方法最大测量误差仅为1.6%,表明了所提方法的有效性和可行性。 展开更多
关键词 构网型变流器 频率变化率 辨识方法
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Optimizing microstructure and mechanical properties of heat-treated Al-Zn-Mg-Cu alloy by indirect hot deformation technology 被引量:3
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作者 ZHANG Quan-da SUN Fu-zhen +1 位作者 LIU Meng LIU Wen-cai 《Journal of Central South University》 SCIE EI CAS CSCD 2022年第11期3544-3556,共13页
In this paper,the indirect thermal tensile experiments of 7075 aluminum alloy including the pre-deformation process at room temperature and the final heat tensile process were carried out,the plastic deformation behav... In this paper,the indirect thermal tensile experiments of 7075 aluminum alloy including the pre-deformation process at room temperature and the final heat tensile process were carried out,the plastic deformation behavior and forming limit of the material in the compound forming process were investigated considering three pre-deformation amounts 4%,9%,14%,two strain rates 0.001 s^(-1),0.01 s^(-1) and four forming temperatures 300℃,350℃,400℃,450℃.In the indirect hot forming process,the material is sensitive to the pre-deformation,strain rate,and forming temperature,when the strain rate is 0.01 s^(-1),the pre-deformation amount is 4%,and the forming temperature is 400℃,respectively,the maximum tensile deformation is 50 mm.Finally,taking the process in which the forming temperature is 450℃ as an example,according to the observation of the microstructure appearance of fracture,the fracture type in the hot forming process was judged as the ductile fracture.By analyzing the microstructure of the specimen treated with the quenching and artificial aging process,the eutectic T(AlZnMgCu)phase and α(Al)matrix formed a network of non-equilibrium alpha binary eutectic. 展开更多
关键词 strain rate PRE-DEformATION forming temperature AA7075 compound tensile
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Effect of P/M value on the preforms and microstructures of spray formed 70Si30Al alloy 被引量:2
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作者 Yanguang Wei Baiqing Xiong Yongan Zhang Hongwei Liu Feng Wang Baohong Zhu 《Journal of University of Science and Technology Beijing》 CSCD 2007年第2期141-145,共5页
A novel 70Si30Al alloy was prepared by the spray forming process for electronic packaging materials. The effect of the ratio of atomization pressure to metal melt mass flux rate (P/M) on the preforms and microstruct... A novel 70Si30Al alloy was prepared by the spray forming process for electronic packaging materials. The effect of the ratio of atomization pressure to metal melt mass flux rate (P/M) on the preforms and microstructures of the spray-deposited 70Si30Al alloy was studied. The results indicate that the PIM value has a considerable influence on the formation of the preforms and the optimal value is in the range of 0.209-0.231 MPa/(kg.min^-1). The microstructure of the spray formed 70Si30AI alloy is fine and homogenous. The primary silicon phases distributing in the aluminum matrix evenly are fine and irregular. The aluminum matrix is divided into two groups: supersaturated α-Al phase or α-Al phase and Al-Si pseudoeutectic phase or Al-Si eutectic phase. 展开更多
关键词 70Si30Al alloy spray forming atomization pressure mass flux rate microstructure
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Improved Correlation for the Volume of Bubble Formed in Air-Water System 被引量:2
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作者 王红一 董峰 +1 位作者 卞聿晨 谭超 《Chinese Journal of Chemical Engineering》 SCIE EI CAS CSCD 2011年第3期529-532,共4页
In order to address the bubble formation and movement in air-water two-phase flow,single bubble rising in stagnant water is experimentally studied by digital image processing.Bubbles are released individually from the... In order to address the bubble formation and movement in air-water two-phase flow,single bubble rising in stagnant water is experimentally studied by digital image processing.Bubbles are released individually from the submerged orifices with different diameters(1.81 mm,2.07 mm,2.98 mm,3.92 mm)at different detachment frequency.Images are recorded by a high-speed video camera and processed by digital image processing technique. The factors impacting the formed volume of bubble are discussed.The experimental results showed that a threshold of gas flow rate(400 mm 3 ·s- 1)divides the bubble formation into two regimes:the constant volume regime and the growing volume regime.Especially for the growing volume regime,the surface tension is taken into account.The bubble volume is consisted of two parts:the surface tension impacting part and the gas volume flow rate impacting part.An improved correlation for bubble volume prediction is developed for the two regimes and better coincidence with the experiment data than the previous models is obtained. 展开更多
关键词 high-speed image bubble formed volume gas volume flow rate gas orifice diameter
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Effects of cooling rate on vermicular graphite percentage in a brake drum produced by onestep cored wire injection
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作者 Yu-shuang Feng Ze-sheng Ji +1 位作者 He-sheng Wu Yong-cheng Feng 《China Foundry》 SCIE CAS 2015年第5期349-353,共5页
In this research, a vermicular graphite cast iron brake drum was produced by cored wire injection in a one-step method. Silica sand and low-density alumina-silicate ceramic were used as molding materials in order to i... In this research, a vermicular graphite cast iron brake drum was produced by cored wire injection in a one-step method. Silica sand and low-density alumina-silicate ceramic were used as molding materials in order to investigate the effect of cooling rate on percentage of vermicular graphite and mechanical properties of the brake drum casting. Several thermocouples were inserted into the casting in the desired positions to measure the temperature change. By means of one-step cored wire injection, the two residual concentrations of Mg and RE were effectively controlled in the ranges of 0.013%-0.017% and 0.019%-0.025%, respectively, which are crucial for the production of vermicular graphite cast iron and the formation of vermicular graphite. In addition, the cooling rate had a significant effect on the vermicular graphite percentage. In the case of the silica mold brake drum casting, there was an obvious difference in the cooling rate with the wall change, leading to a change in vermicular graphite percentage from 70.8% to 90%. In the low-density alumina-silicate ceramic mold casting, no obvious change in temperature was detected by the thermocouples and the percentage of the vermicular graphite was stable at 85%. Therefore, the vermicular graphite cast iron brake drum with a better combination of mechanical properties could be obtained. 展开更多
关键词 cored wire injection in one-step vermicular graphite cast iron cooling rate vermicular graphite percentage brake drum
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Error Estimations, Error Computations, and Convergence Rates in FEM for BVPs
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作者 Karan S. Surana A. D. Joy J. N. Reddy 《Applied Mathematics》 2016年第12期1359-1407,共49页
This paper presents derivation of a priori error estimates and convergence rates of finite element processes for boundary value problems (BVPs) described by self adjoint, non-self adjoint, and nonlinear differential o... This paper presents derivation of a priori error estimates and convergence rates of finite element processes for boundary value problems (BVPs) described by self adjoint, non-self adjoint, and nonlinear differential operators. A posteriori error estimates are discussed in context with local approximations in higher order scalar product spaces. A posteriori error computational framework (without the knowledge of theoretical solution) is presented for all BVPs regardless of the method of approximation employed in constructing the integral form. This enables computations of local errors as well as the global errors in the computed finite element solutions. The two most significant and essential aspects of the research presented in this paper that enable all of the features described above are: 1) ensuring variational consistency of the integral form(s) resulting from the methods of approximation for self adjoint, non-self adjoint, and nonlinear differential operators and 2) choosing local approximations for the elements of a discretization in a subspace of a higher order scalar product space that is minimally conforming, hence ensuring desired global differentiability of the approximations over the discretizations. It is shown that when the theoretical solution of a BVP is analytic, the a priori error estimate (in the asymptotic range, discussed in a later section of the paper) is independent of the method of approximation or the nature of the differential operator provided the resulting integral form is variationally consistent. Thus, the finite element processes utilizing integral forms based on different methods of approximation but resulting in VC integral forms result in the same a priori error estimate and convergence rate. It is shown that a variationally consistent (VC) integral form has best approximation property in some norm, conversely an integral form with best approximation property in some norm is variationally consistent. That is best approximation property of the integral form and the VC of the integral form is equivalent, one cannot exist without the other, hence can be used interchangeably. Dimensional model problems consisting of diffusion equation, convection-diffusion equation, and Burgers equation described by self adjoint, non-self adjoint, and nonlinear differential operators are considered to present extensive numerical studies using Galerkin method with weak form (GM/WF) and least squares process (LSP) to determine computed convergence rates of various error norms and present comparisons with the theoretical convergence rates. 展开更多
关键词 Finite Element Error Estimation Convergence rate A Priori A Posteriori BVP Variationally Consistent Integral form Variationally Inconsistent Integral form Differential Operator Classification SELF-ADJOINT NON-SELF-ADJOINT Nonlinear
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Electron Exchange Between Oxidized and Reduerd Forms of Ferrocene Derivatives
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作者 Chun Ying DUAN Long Gen ZHU Hong Zhe SUN Xiao Zeng YOU Coordination Chemistry Institute of Nanjing University Nanjing,210008 《Chinese Chemical Letters》 SCIE CAS CSCD 1992年第1期57-58,共2页
The rates of electron exchange between oxidized and reduced forms of sev- eral derivatives of ferrocene(1-3)have been measured in acetonitrile by the NMR line h(?)oadening method at 25℃。It was found that the r(?)tes... The rates of electron exchange between oxidized and reduced forms of sev- eral derivatives of ferrocene(1-3)have been measured in acetonitrile by the NMR line h(?)oadening method at 25℃。It was found that the r(?)tes vary with the lengtn of the substitutent chain and the potential of E_(Fe(Ⅲ)/Fe(Ⅱ))in the ferrocene derivatives. 展开更多
关键词 rate Electron Exchange Between Oxidized and Reduerd forms of Ferrocene Derivatives
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