散体移动规律与放矿理论研究.pdf

1 分类号 密级 UDC 学 位 论 文 散体移动规律与放矿理论研究散体移动规律与放矿理论研究 作 者 姓 名 乔乔 登登 攀攀 指导导师姓名 任任 凤凤 玉玉 教教 授授 东北大学资源与土木工程学院东北大学资源与土木工程学院 申请学位级别 博 士博 士 学 科 类 别 工 学工 学 学科专业名称 采矿工程采矿工程 论文提交日期 2003.11 论文答辩日期 2004.10.22 学位授于日期 答辩委员会主席 评 阅 人 东 北 大 学东 北 大 学 2003 年 11 月 2 Ph.D. Dissertation Requirements for the Degree of Doctor of Engineering Study on Flow Law of the Granular Materials under Gravity And Ore-drawing Theory by Qiao Dengpan Supervisor Prof. Ren Fengyu Speciality Mining Engineering School of Civil Engineering and Resources Northeastern University P.R.of China 3 声声 明明 本人声明所呈交的学位论文是在导师的指导下完成的。 论 文中取得的研究成果除加以标注和致谢的地方外， 不包含其他 人已经发表或撰写过的研究成果， 也不包括本人为获得其他学 位而使用过的材料。 与我一同工作的同志对本研究所做的任何 贡献均已在论文中做了明确的说明并表示谢意。 本人签名 日 期 内内 容容 摘摘 要要 4 崩落采矿法在国内外金属矿山应用广泛，并且随着采矿技术的发展和开采深 度的增加其应用比例也在增大。放矿控制是崩落法采矿的核心难题，研究放矿理 论，使之最大限度地接近实际并能有效指导生产，综合降低崩落法开采中的损失 贫化、提高资源回收率是采矿工程师追求的目标。 散体性态的特殊性决定了固体连续介质理论和流体力学理论只能部分适用于 散体。 本论文针对散体的性质和二次松散特点， 建立了重力作用下散体移动流动 微分方程和下沉微分方程，对散体的移动规律做出了数学解析。通过理论推导和 实验分析，首次建立了散体移动体积连续方程和移动体积总流连续方程，二者同 时成立。通过对散体移动微分方程的分析，指出应用抛物线型偏微分方程研究放 矿问题是可行的。当假设散体水平均质且水平各项同性时，上述方程可简化为 无限边界条件下空间问题 2 2 2 2 V y v x v zB z v zzz ∂ ∂ ∂ ∂ − ∂ ∂ 半无限边界条件下空间问题 2 2 2 2 ,V x v zxD y v x v zB z v zzzz ∂ ∂ − ∂ ∂ ∂ ∂ − ∂ ∂ 散体移动体积连续性方程 0 − ∂ ∂ ∂ ∂ ∂ ∂ V z v y v x v z y x 散体移动体积总流连续方程 ∫ Σ 1 1 1 dsvq z dsdzV z ∫∫ Σ 1 0 其中zB----散体水平移动系数，,zxD----散体下沉偏斜系数， V----散体移 动体积变化率。 采场崩落矿岩散体结构非常复杂，且孔隙率很小。放矿过程中散体移动的同 时又伴随着散体本身结构的变化，发生二次松散现象。随着崩落法落矿技术的发 展，崩落矿石的一次松散系数很小甚至仅为1.05，而矿石破碎质量相当好。显然， 此类情况下已不能忽视散体二次松散对放矿效果的影响，否则会造成很大误差。 然而由于实验手段的限制，现有的放矿理论对膨胀散体的移动规律研究甚少。本 论文系统地研究了二次松散影响下的崩落矿岩散体的移动规律，丰富和完善了放 矿随机介质理论。 基于散体随机移动与移动概率分布的一致性关系，提出了散体移动场与概率 场的统一条件即颗粒移动迹线上任意两点横坐标之比等于对应层位的概率分布 方差之比。在假设膨胀散体的移动与二次松散是同时发生且相互独立的物理过程 的基础上，建立了底部及端部放矿时矿岩散体移动概率密度方程，该方程可作为 应用随机介质理论研究崩落法放矿的基础方程。系统地建立了底部放矿及端部放 矿时矿岩散体密度场方程、速度场方程、移动漏斗方程、放出体表面方程、颗粒 移动坐标变换方程。系统地阐述了放出体的过渡关系指出无膨胀散体放出体之 间保持线性过渡关系，放出体表面以等厚度过渡；而膨胀散体放出体之间保持非 线性过渡关系，放出体表面以变厚度过渡。本论文建立的概率密度方程为 5 底部点源放出条件下exp 2 11 ,,, 0 2 n z kz zyxp −         πη η 2 0 2 22 n kz yx − − 端部点源放出条件下exp 2 1 ,,, 0 2 n z kz A zyxp −         πη η 2 0 2 22 n kz yxx − − − 方程中参数包括与散体粘结性有关的流动系数 0 n,散体侧向移动数k，二次松 散系数η，端壁切割系数A，流轴曲线x可依具体端壁条件而定，且各项参数的 物理意义明确，比较全面地考虑了采场硬壁边界因素与散体流动参数对移动规律 的影响。 根据实际放矿口散体流动状态，分析了放出口散体流动速度分布，建立了实 际放出口影响下崩落矿岩散体的移动概率密度方程，给出了放出口影响范围的估 算式，揭示了放矿口对矿岩散体移动规律的影响机理。根据崩落法采场放矿工艺， 系统地研究了底部放矿和端部放矿情况下，放出口尺寸与放出口速度分布对矿岩 散体放出过程中的速度场、密度场、颗粒移动迹线、移动漏斗方程和放出体的影 响，建立了相应的方程，为放出体、残留体、矿岩接触面的移动过程、贫化损失 指标的计算与预测，以及采场合理结构参数的确定奠定了基础。 根据散体移动速度分布确定了矿岩散体有效移动带，并结合放出体方程给出 了无底柱分段崩落法采场结构参数的近似计算公式。 进路间距 SkHL n − 0 2 32 放矿步距 [] 2 2 3 1 , 0 RHkifR n − θ 式中 H分段高度，S进路宽度， R 放出口沿进路方向散体有效流动范围， θ端壁倾角，i散体自然安息角, 端壁阻尼系数，其它符号意义同前。 本论文由于系统地研究了二次松散的影响，理论方程与实验结果符合较好， 使放矿理论与实际更为接近，可以预见本论文研究成果在指导崩落法采场设计和 控制放矿中有很高的推广应用价值。 本论文建立的散体移动微分方程在相应的简化条件下的解与根据随机介质放 矿理论建立的速度场方程是相同的，在理论上取得了一致，不仅形成了系统的放 矿理论体系，而且二者的有效结合可以进一步研究复杂散体系统。对由近地表结 构工程引起的地表沉降问题，特别是对研究近地表岩土层内部各处的大位移问题 有很高的应用价值。 关键词关键词散体移动微分方程 体积连续方程 放矿随机介质理论 松散系数 放出体 放出口 端壁 结构参数 6 ABSTRACT Stope-caving system has been widely used in the world, and with the developing of mining technology and the increasing of mining depth, the proportion of using it is getting increase. Ore-drawing control is the difficult core problem of stope-caving system. It is the goal of mining engineers to study ore-drawing theory and take it into directing discharging operation so effectively that the dilution and the ore loss can be lowed and the recovery of resources can be increased as much as possible. On account of the special quality of granular materials, solid continuum medium theory and hydrodynamics theory can be partly applicable to granular materials. On the base of tests and analysis, the author deduced motion flow differential equation and subsidence differential equation of loose body under gravity, and gave mathematical resolution to the flow law of loose body in this thesis. The author firstly set up volume continuum equation and volume total-flow continuum equation of motion loose body that two equations stand in existence together. By analyzing the motion differential equation, the author pointed out it is feasible to use parabolic differential equation to study ore drawing. Suppose the loose body is homogeneous and isotropic in horizontal direction, the equations above can be simplified as follows Space problem of infinite boundary condition 2 2 2 2 V y v x v zB z v zzz ∂ ∂ ∂ ∂ − ∂ ∂ Space problem of semi-infinite boundary condition 2 2 2 2 ,V x v zxD y v x v zB z v zzzz ∂ ∂ − ∂ ∂ ∂ ∂ − ∂ ∂ Volume continuum equation of motion loose body 0 − ∂ ∂ ∂ ∂ ∂ ∂ V z v y v x v z y x Volume total-flow continuum equation of motion loose body ∫ Σ 1 1 1 dsvq z dsdzV z ∫∫ Σ 1 0 Where zBhorizontal motion coefficient of loose body, ,zxDsubsidence deviation coefficient of loose body, Vvolume change rate of motion loose body. The blasted ore-rock loose body is very complicated in inner structure, composition and distribution, which only completed the first loose. Drawing ore is the process of bottom-offloading, and there must be the second loose phenomenon resulted by structure transmogrification of loose body during the drawing. With the developing of blasting technique, the first loose coefficient is very low even to 1.05, but the crushed quality of ore is very good. Obviously, on this case the influence of the second loose to drawing effect should be taken into account, otherwise, there will be greater error. However, nearly all of the existent theories about studying the law of the loose body flow rarely take it into consideration for the limit of the experiment, which makes the ed models far from actualities and results far from those observed in model and 7 full-scale experiments. This thesis, therefore, focused on the flow law of blasted ore-rock mass affected by re-loosening under gravity, enriched and improved the stochastic theory of ore-drawing. Based on the uniity relation of the random motion of granules and the distribution of movement probability, the author pointed out the uni condition using stochastic theory to study ore-drawing, that is the rate of the x-coordinates of any two points on any movement trajectories of granules equals the rate of variances of corresponding levels. Otherwise, the movement probability field can’t coincide with the movement and velocity field of loose body. On the base of supposing the motioning of loose body and the re-loosening is a physical process of mutual independence and simultaneity, the author put forward the movement probability density equations of non-dilatant and dilatant loose materials as discharging downwards under gravity, which can be thought as the basic equation to study the flow law of granular mass. The author systematically set up the density equations, the velocity equations, the discharging funnel equations, the drawn-body equations and the coordinates equations of moving granules on the condition of bottom discharging and solid wall discharging. The author concluded the transition relations of the drawn bodies is that linear transition is remained in non-dilatant materials dischargingeven thickness transition relation, but non-linear in dilatant materialsuneven thickness transition relation. Where the probability density equations of movement as follows Bottom dot-opening discharging exp 2 11 ,,, 0 2n z kz zyxp −         πη η 2 0 2 22 n kz yx − − Solid wall dot-opening discharging exp 2 1 ,,, 0 2n z kz A zyxp −         πη η 2 0 2 22 n kz yxx − − − The parameters in those equations above are 0 nflow coefficient in relation to the caking property of loose body, kthe value of side movement, ηthe second loose coefficient, Ainfluence coefficient of solid wall, xthe flow axis depended on specific solid wall. In brief, the effects of solid wall and flow parameters to the movement of loose body are comprehensively taken into account in this thesis, as well as the meanings of all parameters in physics are clear in equations above. According to the characteristic pattern of loose body flow and flow velocity distribution at discharging opening, the author deduced the probability density equations of blasted ore-rock mass movement downwards under gravity affected by the actual discharging opening, got the estimation ula about the influence high of the outlet, and uncovered the influence mechanics to blasted ore-rock mass movement downwards. By taking the drawing way, size of the discharging opening and velocity distributions into consideration on the condition of bottom discharging and solid wall discharging, the author systematically set up the velocity equations, density equations, discharging funnel equations, drawn body equations and movement trajectory functions of granules in this thesis, so that the motion processes of drawn body, remnant and ore-rock 8 interface can be calculated, and the ore loss and dilution can be pre-estimated out as accurate as possible after stope structure is given. According to the motion velocity distribution of caved materials under gravity and drawn body equation, the author put forward the effective motion zone in motion field, as well as set up the approximate expressions of stope structure parameters of sublevel caving as follows Space length between drills SkHL n − 0 2 32 Drawing pace [] 2 2 3 1 , 0 RHkifR n − θ The parameters in two expressions above are Hsublevel height, Sdrill width, Reffective flow width along drill of discharging opening, θinclination of the solid wall in relation to the horizontal, damp coefficient of solid wall, iangle of repose of loose body, kthe value of side movement, 0 n flow coefficient in relation to the caking property of loose body. In conclusion, because systematically studied the influence of the second loose in this thesis as well as the theoretical equations fit the model-scale experimental results very well, it can be foreseen that the research fruits in the thesis have better practicability in designing and directing drawing ore control. Furthermore, on the supposing the loose body is homogeneous and isotropic in horizontal direction, the solutions of the motion differential equation of loose body in this thesis accord with the velocity equations set by stochastic theory of ore drawing, so that the research ed not only a newer system of ore drawing, but also a available to study complicated granular materials system, which can be used to study surface subsidence resulted by underground constructional engineering, especially to large displacement of everywhere in rock-soil stratum near surface. Key Words Motion differential equation of loose body Volume continuum equation Stochastic theory of ore drawing Loose coefficient Drawn-body Discharging opening Solid wall Structural parameters 9 目 录 声明⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯Ⅰ 内容摘要⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯Ⅱ ABSTRACTABSTRACT⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯Ⅳ 第一章 放矿理论研究概况第一章 放矿理论研究概况⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯1 1.1 引 言⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯1 1.2 放矿理论研究的目的意义 ⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯1 1.3 放矿理论的研究现状 ⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯2 1.3.1 随机介质放矿理论的研究现状⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯2 1.3.2 椭球体放矿理论的研究现状⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯7 1.3.3 类椭球体放矿理论的研究现状⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯8 1.4.放矿计算机仿真研究现状 ⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯9 1.5.崩落矿岩散体的流动性研究⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯11 1.6.力学在放矿中的应用⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯12 1.6.1 散体静力学在放矿中的应用⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯12 1.6.2 散体动力学在放矿中的应用⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯12 1.6.3 流体力学在放矿中的应用⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯13 1.7.放矿理论研究中存在的问题与研究方向⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯13 1.8.放矿理论研究的前景⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯15 第二章第二章 散体移动规律的数学解析散体移动规律的数学解析⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯16 2.1 散体移动的连续性假设⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯16 2.2 散体移动的连续性方程⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯17 2.2.1 质量连续方程⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯17 2.2.2 体积连续方程⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯19 2.2.2.1 移动散体体积总流连续方程⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯22 2.2.2.2 散体移动体积连续方程⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯23 2.3 散体移动微分方程⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯25 2.3.1 散体移动场⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯25 2.3.1.1 垂直方向移动⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯26 2.3.1.2 水平方向移动⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯27 2.3.1.3 水平移动距离 L 与下沉量 W 的关系⋯⋯⋯⋯⋯⋯⋯⋯28 2.3.2 散体移动微分方程⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯34 2.3.3 硬壁影响下散体移动微分方程⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯35 2.4 散体下沉微分方程⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯37 2.5 微分方程求解与分析⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯38 2.6 散体移动微分方程与放矿随机介质理论⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯41 2.7 本章小结⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯42 10 第三章 底部放矿随机介质理论方程第三章 底部放矿随机介质理论方程⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯44 3.1 概 述⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯44 3.2 底部放矿时矿岩散体移动概率方程⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯44 3.2.1 基本假设⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯44 3.2.2 无膨胀散体移动概率密度方程⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯44 3.2.2.1 无膨胀散体移动模型⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯44 3.2.2.2 递补概率赋值⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯45 3.2.2.3 方差的合理表达式⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯47 3.2.2.4 无膨胀散体移动概率密度方程⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯49 3.2.3 膨胀散体移动概率密度方程⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯49 3.3 移动散体密度场⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯53 3.3.1 移动散体密度场⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯53 3.3.2 等密度体⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯54 3.4 散体移动速度场及其检验⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯55 3.4.1 无膨胀散体移动速度场⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯55 3.4.2 膨胀散体移动速度场⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯57 3.4.3 散体移动连续性检验⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯58 3.4.3.1 无膨胀散体速度场检验⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯58 3.4.3.2 膨胀散体速度场检验⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯58 3.4.4 速度场与运动微分方程⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯59 3.4.5 等速体⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯59 3.4.6 散体移动场划分⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯60 3.4.7 放矿中散体移动范围的近似确定⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯63 3.5 放出漏斗⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯64 3.5.1 无膨胀散体放出漏斗⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯64 3.5.2 膨胀散体放出漏斗⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯65 3.6 放出体⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯66 3.6.1 放出体母线方程⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯66 3.6.2 放出体形态⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯68 3.6.3 降落漏斗体积、放出体体积与放出量的关系⋯⋯⋯⋯⋯⋯⋯⋯70 3.7 底部放矿模拟实验⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯71 3.8 放出体的过渡关系⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯75 3.8.1 放出体表面过渡关系⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯75 3.8.2 放出体与等速体等密度体关系⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯76 3.8.3 放出体表面厚度过渡关系⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯77 3.9 散体颗粒点的移动方程⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯79 3.9.1 颗粒移动坐标的正、逆变换方程⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯79 3.9.2 归零量与坐标变换⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯80 3.10 本章小结⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯81 11 第四章 端部放矿随机介质理论方程第四章 端部放矿随机介质理论方程⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯92 4.1 概述⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯92 4.2 端壁切割条件下崩落矿岩散体移动概率方程⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯93 4.3 端壁前倾条件下崩落矿岩散体移动规律⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯96 4.3.1 概率密度方程⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯96 4.3.2 颗粒移动迹线⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯98 4.3.3 速度场⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯98 4.3.3.1 无膨胀散体移动速度场⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯98 4.3.3.2 膨胀散体移动速度场⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯99 4.3.3.3 速度场检验 ⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯100 4.3.3.4 散体速度场与运动微分方程 ⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯100 4.3.3.5 膨胀散体等速体 ⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯101 4.3.4 密度场⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯102 4.3.5 移动漏斗⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯103 4.3.6 放出体⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯104 4.3.6.1 放出体方程⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯104 4.3.6.2 放出体形态⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯106 4.3.7 颗粒移动坐标变换方程⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯106 4.4 端壁直立条件下崩落矿岩散体移动规律⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯108 4.4.1 移动概率密度方程⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯108 4.4.2 颗粒移动迹线⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯109 4.4.3 速度场⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯110 4.4.3.1 膨胀散体速度场⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯110 4.4.3.2 速度场检验⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯110 4.4.3.3 等速体⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯111 4.4.4 密度场⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯112 4.4.5 移动漏斗⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯112 4.4.6 放出