In this paper, we use a geometric identity in the n-dimensional Euclidean space En and give the further improveme nt of Klamkin inequality in the space En.
The problem on the geometrc inequalities involving an n-dimensional simplex and its inscribed simplex is studied. An inequality is established, which reveals that the difference between the squared circumradius of the...The problem on the geometrc inequalities involving an n-dimensional simplex and its inscribed simplex is studied. An inequality is established, which reveals that the difference between the squared circumradius of the n-dimensional simplex and the squared distance between its circumcenter and barycenter times the squared circumradius of its inscribed simplex is not less than the 2(n-1)th power of n times its squared inradius, and is equal to when the simplex is regular and its inscribed siplex is a tangent point one. Deduction from this inequality reaches a generalization of n-dimensional Euler inequality indicating that the circumradius of the simplex is not less than the n-fold inradius. Another inequality is derived to present the relationship between the circumradius of the n-dimensional simplex and the circumradius and inradius of its pedal simplex.展开更多
The paper presents the next step within multiyear fruitless efforts of the author to overcome the absurd situation in boiling heat transfer research. The focus is made on the problem of the characteristic length of th...The paper presents the next step within multiyear fruitless efforts of the author to overcome the absurd situation in boiling heat transfer research. The focus is made on the problem of the characteristic length of the process most clearly exhibiting the consequences of half a century ignoring the basic MTD （Model ＂theater of director＂）, the UC （Universal correlation） and some other boiling fundamentals. Echoing control of boiling heat transfer by nucleation, the MTD-UC identifies universal characteristic length, the AER （Average effective radius） of nucleation sites, equally workable at the macro- and microscale. Inefficiency of the generally accepted, so called MTA （theater of actors） is particularly pronounced just in the confusion with the characteristic length. Traditional and potential candidates, departure diameter of vapor bubble and transverse internal size of the channel hardly can be adjusted to independence of developed boiling HTC on mass acceleration, subcooling, liquid convection and the heating surface geometry. At the same time, even such a problem has not prevented many authors to develop tens or even hundreds of helpless MTA-based correlations. The ignoring the MTD-UC-AER has also led to the incompleteness of the standard boiling heat transfer experiment, which is usually done without studying nucleation sites （there are available only very few comprehensive experimental works including the data on the AER）. The only exception was made for the problem of boiling heat transfer enhancement： over the past decades enhanced boiling surfaces were developed in direct accordance with the principle defined by the MTD-UC （just through the AER）. Another thing is that the basic role of the MTD-UC-AER in substantial progress of the relevant R＆D activities passed over in silence in the corresponding publications. Enviable unity and coherence of heat transfer community in preventing real scientific debate on the problem is also remarked.展开更多
As one kind of key lightweight components with enormous quantities and diversities, the bent tubular parts have attracted in- creasing applications in aerospace, automobile, etc. Thus, how the inevitable springback be...As one kind of key lightweight components with enormous quantities and diversities, the bent tubular parts have attracted in- creasing applications in aerospace, automobile, etc. Thus, how the inevitable springback behaves under different bending specifications should be fully addressed to efficiently achieve the precision forming of various bent tubes. Taking the medium strength thin-walled 6061-T4 Al-alloy tube as the objective, via the deformation theory of plasticity, explicit/implicit FE method and experimental approaches, we explored and clarified the nonlinear springback rules of the tubes and corresponding mechanisms in universal rotary draw bending regarding angular springback and radius growth by deliberately changing the tube diameter D and wall thickness t. The geometry dependent springback behaviors of thin-walled tube upon cold bending are thus revealed: 1) With the increasing of D, the tangent tensile strain increases and the proportional coefficient decreases, which causes the angular springback to decrease, while the radius springback increases due to the larger bending radius. 2) With the increasing of t, the tangent tensile strain decreases and the proportional coefficient increases, resulting in the increase of both angular springback and radius springback. 3) Under the same D/t, the angular springback varies little, while the radius springback increases with the larger diameter D. 4) The D/t can be used as a reasonable nondimensional index to evaluate the springback angle; as to the radius growth, the individual effects of the D and t should be considered. 5) The verification of the above results was conducted by experiments and analytical analysis.展开更多
文摘In this paper, we use a geometric identity in the n-dimensional Euclidean space En and give the further improveme nt of Klamkin inequality in the space En.
文摘The problem on the geometrc inequalities involving an n-dimensional simplex and its inscribed simplex is studied. An inequality is established, which reveals that the difference between the squared circumradius of the n-dimensional simplex and the squared distance between its circumcenter and barycenter times the squared circumradius of its inscribed simplex is not less than the 2(n-1)th power of n times its squared inradius, and is equal to when the simplex is regular and its inscribed siplex is a tangent point one. Deduction from this inequality reaches a generalization of n-dimensional Euler inequality indicating that the circumradius of the simplex is not less than the n-fold inradius. Another inequality is derived to present the relationship between the circumradius of the n-dimensional simplex and the circumradius and inradius of its pedal simplex.
文摘The paper presents the next step within multiyear fruitless efforts of the author to overcome the absurd situation in boiling heat transfer research. The focus is made on the problem of the characteristic length of the process most clearly exhibiting the consequences of half a century ignoring the basic MTD （Model ＂theater of director＂）, the UC （Universal correlation） and some other boiling fundamentals. Echoing control of boiling heat transfer by nucleation, the MTD-UC identifies universal characteristic length, the AER （Average effective radius） of nucleation sites, equally workable at the macro- and microscale. Inefficiency of the generally accepted, so called MTA （theater of actors） is particularly pronounced just in the confusion with the characteristic length. Traditional and potential candidates, departure diameter of vapor bubble and transverse internal size of the channel hardly can be adjusted to independence of developed boiling HTC on mass acceleration, subcooling, liquid convection and the heating surface geometry. At the same time, even such a problem has not prevented many authors to develop tens or even hundreds of helpless MTA-based correlations. The ignoring the MTD-UC-AER has also led to the incompleteness of the standard boiling heat transfer experiment, which is usually done without studying nucleation sites （there are available only very few comprehensive experimental works including the data on the AER）. The only exception was made for the problem of boiling heat transfer enhancement： over the past decades enhanced boiling surfaces were developed in direct accordance with the principle defined by the MTD-UC （just through the AER）. Another thing is that the basic role of the MTD-UC-AER in substantial progress of the relevant R＆D activities passed over in silence in the corresponding publications. Enviable unity and coherence of heat transfer community in preventing real scientific debate on the problem is also remarked.
基金supported by the National Natural Science Foundation of China (Grant No. 50905144)Program for New Century Excellent Talentsin University+2 种基金the fund of the State Key Laboratory of Solidification Processing in NWPUthe Natural Science Basic Research Plan in Shaanxi Province (Grant No. 2011JQ6004)the 111 Project (Grant No.B08040)
文摘As one kind of key lightweight components with enormous quantities and diversities, the bent tubular parts have attracted in- creasing applications in aerospace, automobile, etc. Thus, how the inevitable springback behaves under different bending specifications should be fully addressed to efficiently achieve the precision forming of various bent tubes. Taking the medium strength thin-walled 6061-T4 Al-alloy tube as the objective, via the deformation theory of plasticity, explicit/implicit FE method and experimental approaches, we explored and clarified the nonlinear springback rules of the tubes and corresponding mechanisms in universal rotary draw bending regarding angular springback and radius growth by deliberately changing the tube diameter D and wall thickness t. The geometry dependent springback behaviors of thin-walled tube upon cold bending are thus revealed: 1) With the increasing of D, the tangent tensile strain increases and the proportional coefficient decreases, which causes the angular springback to decrease, while the radius springback increases due to the larger bending radius. 2) With the increasing of t, the tangent tensile strain decreases and the proportional coefficient increases, resulting in the increase of both angular springback and radius springback. 3) Under the same D/t, the angular springback varies little, while the radius springback increases with the larger diameter D. 4) The D/t can be used as a reasonable nondimensional index to evaluate the springback angle; as to the radius growth, the individual effects of the D and t should be considered. 5) The verification of the above results was conducted by experiments and analytical analysis.
