浅谈小学数学教学中渗透数形结合思想

数形结合具有直观性、形象性的双重意义。它能将代数与几何结合起来,有助于降低学习数学的难度,培养学生的数学素养。在此基础上,数学与几何的结合,小学数学教师必须认识到数形结合的重要性和应用的意义,在课堂教学中采用有效的教学方法,运用数形结合,提高学生学习数学的兴趣,提高课堂教学质量。

数感——数学学习的基石——浅谈数感在教学中的有机渗透

数感指的是学生对数字、数字间关系,以及它们如何受到外界运动影响的一种直观感觉,它可以表现为对数字的敏感度、反应力和鉴赏性。为了更好地培养学生的数感,教师需要在日常教学中通过以生活实例为数感背景、以相互比较为数感认知、以动手操作为数感实践、以解决问题为数感内化这几种方式去帮助学生逐步建立起对数的领悟能力,从而更好地夯实数学学习的基石。

多孔质径向气体静压轴承运动稳定性分析

多孔质轴承在高速运转条件下会出现涡动失稳现象,使轴承的动态稳定性下降。利用线性扰动法对多孔质静压轴承中气体润滑的连续方程求解,得到动态下轴承稳定的临界质量———系统保持稳定的最小质量,并以此为稳定性判据得出轴承参数对稳定性的影响,即轴承的供气压力、渗透数、长径比越大,稳定区域越小,为今后的多孔质轴承设计提供了参数的优化范围。

The Unsaturated Hydraulic Parameters for Aeolian Sand

The water characteristic curve for aeolian sand in two processes of wetting and drying was obtained by the negative water column technique.The values of fitting parameters were calculated according to Van Genuchten formula and the parameters that characterized the prosperities of aeolian sand such as the unsaturated infiltration coefficient and specific water capacity were obtained.The results showed that the water characteristic curve for aeolian sand in wetting process had greater hysteresis quality than ...

Seepage characteristics of a fractured silty mudstone under different confining pressures and temperatures

To investigate the influence of confining pressures and temperatures on the seepage characteristics of fractured rocks, seepage tests were conducted on a fractured silty mudstone using a self-developed experimental system, and the effects of different factors on coefficient of permeability were discussed. The results showed that the increasing confining pressure will gradually decrease the coefficient of permeability, and this process is divided into two stages: 1) the fast decrease stage, which corresponds to a confining pressure less than 30 kPa, and 2) the slow decrease stage, which corresponds to a confining pressure larger than 30 kPa. Unlike confining pressure, an increase in temperature will increase the coefficient of permeability. It is noted that fracture surface roughness will also affect the variation of coefficient of permeability to a certain extent. Among the three examined factors, the effect of confining pressure increases is dominant on fracture permeability coefficient. The relationship between the confining pressure and coefficient of permeability can be quantified by an exponential function.

Self-contained in situ sediment acoustic measurement system based on hydraulic driving penetration

A novel self-contained in situ sediment acoustic measurement system based on hydraulic driving penetration is proposed to solve the problem of large disturbances to sediments of the in situ equipments already in existence. By using a hydraulic driving device, the system drives four acoustic probes into sediments at an even speed, and this decreases disturbances to sediments introduced by the penetration of acoustic probes. By means of the special design of the central control unit, the system can work full-automatically and the data are stored self-containedly, and this avoids the requirement of real-time remote controlling from the ship. Its operating water depth, measuring depth and measuring frequency is 500m, 1.0m and 30kHz respectively. A set of in situ sound speeds and attenuation coefficients of sediments are obtained at 40 stations using the system. The results confirm that the data obtained by the in situ sediment acoustic system are accurate and credible.

Unsaturated expansive soil fissure characteristics combined with engineering behaviors

The relationship among the surface fissure ratio, moisture content, seepage coefficient and deformation modulus of field unsaturated expansive soil in Nanning, Guangxi Province, China, was obtained by a direct or indirect method. Digital images of expansive soil of the surface fissure with different moisture contents were analyzed with the binarization statistic method. In addition, the fissure fractal dimension was computed with a self-compiled program. Combined with in situ seepage and loading plate tests, the relationship among the surface fissure ratio, moisture content, seepage coefficient and deformation modulus was initially established. The surface fissure ratio and moisture content show a linear relation, ＂y=-0.019 1x＋1.028 5＂ for rufous expansive soil and ＂y=-0.07 1x＋2.610 5＂ for grey expansive soil. Soil initial seepage coefficient and surface fissure ratio show a power function relation, ＂y=1× 10^-9exp（15.472x）＂ for rufous expansive soil and ＂y=5× 10^-7exp（4.209 6x）＂ for grey expansive soil. Grey expansive soil deformation modulus and surface fissure ratio show a power fimction relation of ＂y=3.935 7exp（0.993 6x）＂. Based on the binarization and fractal dimension methods, the results show that the surface fissure statistics can depict the fissure distribution in the view of two dimensions. And the evolvement behaviors of permeability and the deformation modulus can indirectly describe the developing state of the fissure. The analysis reflects that the engineering behaviors of unsaturated expansive soil are objectively influenced by fissure.

Retention of clay-solidified grouting curtain to Cd^(2+), Pb^(2+) and Hg^(2+) in landfill of municipal solid waste

The effects of components and their ratio of grouts on anti-seepage capability of clay-solidified grouting curtain and its permeability of heavy metal cations were investigated by permeating experiments, using reactive solute transport model to study the permeation of heavy metals (Cd2+, Pb2+ and Hg2+). The study of permeating for different mixture ratios of cement and clay indicates that hydraulic conductivity of clay-solidified grouting curtain with different ratios of solid to liquid or with the same ratio of solid to liquid but with different ratios of cement to clay is changed. The laboratory simulation test results also show that precipitates produced in heavy metal cation migration process in curtain block up water flowing passage which makes the hydraulic conductivity of the solution-permeated curtain decrease with the leakage time. The permeation velocities for different heavy metal cations vary with ionic concentration, exchange capacity and ion radius etc. The test results indicate that the permeation rapidity order of heavy metals cations in clay-solidified grouting curtain is Hg2+>Pb2+ in the same experimental circumstance. In addition, permeability for different mixture ratios and antisepsis capabilities of clay-solidified grouting curtain were studied in tests.

A numerical simulation of seepage structure surface and its feasibility

According to Cubic law and incompressible fluid law of mass conservation, the seepage character of the fracture surface was simulated with the simulation method of fractal theory and random Brown function. Furthermore, the permeability coefficient of the single fracture was obtained. In order to test the stability of the method, 500 simulations were conducted on each different fractal dimension. The simulated permeability coefficient was analyzed in probability density distribution and probability cumulative distribution statistics. Statistics showed that the discrete degree of the permeability coefficient increases with the increase of the fractal dimension. And the calculation result has better stability when the fractal dimension value is relatively small. According to the Bayes theory, the characteristic index of the permeability coefficient on fractal dimension P（Dfi｜ Ri） is established. The index, P（Dfi｜ Ri）, shows that when the simulated permeability coefficient is relatively large, it can clearly represent the fractal dimension of the structure surface, the probability is 82%. The calculated results of the characteristic index verify the feasibility of the method.

Mathematical Model of Gas Permeation Through PTFE Porous Membrane and the Effect of Membrane Pore Structure

Membrane-based separation processes are new technology combined membrane separation with conventional separation. Hydrophobic porous membranes are often used in these processes. The structure of hydrophobic porous membrane has significant effect on mass transfer process. The permeabilities of five kinds of gas, He, N2, O2, CO2 and water vapor, across six polytetrafluoroethylene(PTFE) flat membranes were tested experimentally. Results indicated that the greater the membrane mean pore size and the wider the pore size distribution are, the higher the gas permeability. A gas permeation model, including the effects of membrane structure parameter and gas properties, was established. A comprehensive characteristic parameter (including porosity, thickness and tortuosity) was found more effective to express the influence of membrane structure in gas permeation process. The predicted permeation coefficients were in good agreement with experimental data.

A method of determining the permeability coefficient of coal seam based on the permeability of loaded coal

This study developed the equipment for thermo-fluid–solid coupling of methane-containing coal, and investigated the seepage character of loaded coal under different working conditions. Regarding the effective pressure as a variable, the variation characteristics of the gas permeability of loaded methane-containing coal has been studied under the conditions of different confining pressures and pore pressures. The qualitative and quantitative relationship between effective stress and permeability of loaded methane-containing coal has been established, considering the adsorption of deformation, amount of pore gas compression and temperature variation. The results show that the permeability of coal samples decreases along with the increasing effective stress. Based on the Darcy law, the correlation equation between the effective stress and permeability coefficient of coal seam has been established by combining the permeability coefficient of loaded coal and effective stress. On the basis of experimental data, this equation is used for calculation, and the results are in accordance with the measured gas permeability coefficient of coal seam. In conclusion, this method can be accurate and convenient to determine the gas permeability coefficient of coal seam, and provide evidence for forecasting that of the deep coal seam.

An estimation of the penetration rate of rotary drills using the Specific Rock Mass Drillability index

The main purpose of the present study was to provide a practical, convenient drillability prediction model based on rock mass characteristics, geological sampling from blast holes, and drill operational factors. Empirical equations that predict drill penetration rate have been developed using statistical analyses of data from the Sarcheshmeh Copper Mine. Seven parameters of the rock or rock mass, including uniaxial compressive strength （UCS） of the rock, Schmidt hammer hardness value, quartz content, fragment size （dso）, alteration, and joint dip, are included in the model along with two operational parameters of the rotary drill, bit rotational speed and thrust. These parameters were used to predict values of the newly developed Specific Rock Mass Drillability （SRMD） index. Comparing measured SRMD values to those pre- dicted by the multi-parameter linear, or nonlinear, regression models showed good agreement. The cor- relation coefficients were 0.82 and 0.81. resoectively.

Rules for confidence intervals of permeability coefficients for water flow in over-broken rock mass

Based on the steady-state seepage method, we used the Mechanical Testing and Simulation 815.02 System and a self-designed seepage instrument for over-broken stone to measure seepage properties of water flows in three types of crushed rock samples. Three methods of confidence interval in describing permeability coefficients are presented: the secure interval, the calculated interval and the systemic interval. The lower bound of the secure interval can be applied to water-inrush and the upper bound can solve the problem of connectivity. For the calculated interval, as the axial pressure increases, the length of confidence interval is shortened and the upper and lower bounds are reduced. For the systemic interval, the length of its confidence interval, as well as the upper and lower bounds, clearly vary under low axial pressure but are fairly similar under high axial pressure. These three methods provide useful information and references for analyzing the permeability coefficient of over-broken rock.

Effect of granite gravel content on improved granular mixtures as railway subgrade fillings

The improved granular mixtures are widely used as the fillings of railway 8ubgrade, and in order to investigate the effect of coarse grain content on granular mixtures, a series of field tests were conducted. The experimental results indicate that the permeability coefficient increases significantly with the increment of granite gravel content, especially in the range of 60%-70%. Thcrc exists a coarse grain content limit defined as 53%-58.5% to reform the permeable granular skeleton. Beyond this limit, the permeable granular skeleton is efficiently formed, and the macro pores between the separate gravels are partially filled, which is the explanation lbr the permeability increase. The investigations indicate the subgrade resistance modulus （ks0, Ev2, and Evd） depends on the granite gravel content, and the resistance modulus increases significantly beyond granite gravel content of 50%. The skeletons of granitc gravel clayey sand mixture change in the long-term deformation objected to the train-induced dynamic load, which involves three main repeated and circular deformation stages. Generally, the long-time deformation is explained as the gravel crushing and filling the internal porous space with crushed gravel fragments. Through these investigations, the C40-G60 or C30-G70 is recommended as an optimum soil mixture for the good permeability and high resistance modulus.

Microscopic analysis of saturated soft clay in Pearl River Delta

A series of researches were carried out for the soil samples in the Pearl River Delta under the action of consolidation loads,such as the quantitative analyses of the pore scale,shape and size distributions of micro-structure units,with an environmental scanning electron microscope (ESEM),a mercury intrusion analyzer and a mineral diffractometer. The experimental results show that the consolidation pressures remarkably change the pore sizes and distribution characteristics of the silt,thus changing its compressibility and permeability. This can be proved by the fact that,in the earlier stage with a consolidation pressure of p<200 kPa,the pore sizes are greater and the compressibility and permeability coefficients are larger. However,they rapidly decrease with the increase in consolidation pressure. And in the later stage with a consolidation pressure of p>200 kPa,the pore sizes are smaller and the compressibility and permeability coefficients are less. Therefore,the empirical formulas of compression coefficient and permeability coefficient vs consolidation load and average pore diameter are deduced.

Nonlinear flow numerical simulation of low-permeability reservoir

A nonlinear flow reservoir mathematical model was established based on the flow characteristic of low-permeability reservoir.The well-grid equations were deduced and the dimensionless permeability coefficient was introduced to describe the permeability variation of nonlinear flow.The nonlinear flow numerical simulation program was compiled based on black-oil model.A quarter of five-spot well unit was simulated to study the effect of nonlinear flow on the exploitation of low-permeability reservoir.The comprehensive comparison and analysis of the simulation results of Darcy flow,quasi-linear flow and nonlinear flow were provided.The dimensionless permeability coefficient distribution was gained to describe the nonlinear flow degree.The result shows that compared with the results of Darcy flow,when considering nonlinear flow,the oil production is low,and production decline is rapid.The fluid flow in reservoir consumes more driving energy,which reduces the water flooding efficiency.Darcy flow model overstates the reservoir flow capability,and quasi-linear flow model overstates the reservoir flow resistance.The flow ability of the formation near the well and artificial fracture is strong while the flow ability of the formation far away from the main streamline is weak.The nonlinear flow area is much larger than that of quasi-linear flow during the fluid flow in low-permeability reservoir.The water propelling speed of nonlinear flow is greatly slower than that of Darcy flow in the vertical direction of artificial fracture,and the nonlinear flow should be taken into account in the well pattern arrangement of low-permeability reservoir.

Experimental Study of Seepage Properties of Non-Darcy Flow in Granular Coal Gangues

By using the steady-state seepage method, a patent seepage device together with the MTS815.02 Rock Mechanics Test System is used to test the seepage properties of non-Darcy flow in a granular gangue with five different grain sizes during the compaction. The experimental results show that the seepage properties are not only related to the stress or displacement level, but also to the grain size, the pore structure of the granular gangue, and the current porosity The permeability and the non-Darcy flow coefficient can be fitted respectively by the cubic polynomials and the power functions of the porosity, Formally, the flow in granular gangue satisfies the Forchheimer＇s binomial flow, but under the great axial and confining pressure and owing to the grain＇s crushing, the flow in granular gangues is different from that in rock-fills which are naturallv oiled un. As a result, the non-Darer flow coefficient may be negative.

Seepage Anisotropy of Heterogeneous Body

Ore dumps are heterogeneous bodies with anisotropic seepage characteristics because of the ore segregation. In an indoor experiment,a dump was constructed with three strata,where the horizontal and vertical seepage experi-ments were carried out. Horizontals flow are regarded as phreatic plan flows without penetration. Its seepage law sati-fies the Dupuit equation. With parallel lay seepage model,the equivalent seepage coefficient in the horizontal flow was obtained and was equivalent to the weighted mean of the seepage coefficient of each stratum. An unsaturated flow ap-peared in the vertical experiment,with a hydraulic gradient of 1. The vertical flow was equivalent to the seepage model that moved in vertical bedding; its equivalent seepage coefficient depended on the stratum with the minimum seepage coefficient. That the experiment showed clear anisotropy in a heterogeneous body was obvious with an anisotropic co-efficient between 63 and 155,which is 25 to 100 times larger than that of a homogeneous body.

Influences affecting the soil-water characteristic curve

The soil-water characteristic curve (SWCC) is the primary partially saturated soil information as its behavior and properties can be derived from it. Although there have been many studies of unsaturated soils and the SWCC, there is still no combined constitutive model that can simulate soil characteristics accurately. In cases when hydraulic hysteresis is dominant (e.g. under cyclic loading) it is particularly important to use the SWCC. In the past decades, several mathematical expressions have been proposed to model the curve. There are various influences on the SWCC as a source of information, so the curves obtained from conventional tests often cannot be directly applied; and the mathematical expressions from one scenario cannot be used to simulate another situation. The effects of void ratio, initial water content, stress state and high suction were studied in this work revealing that water content and stress state are more important than the other effects; but that the influences tend to decrease when suction increases. The van Genuchten model was modified to simulate better the changes in the degree of saturation at low values of suction. Predictions were compared with experimental results to determine the simulation capability of the model.

Active earth pressure acting on retaining wall considering anisotropic seepage effect

This paper presents a general solution for active earth pressure acting on a vertical retaining wall with a drainage system along the soil-structure interface. The backfill has a horizontal surface and is composed of cohesionless and fully saturated sand with anisotropic permeability along the vertical and horizontal directions. The extremely unfavourable seepage flow on the back of the retaining wall due to heavy rainfall or other causes will dramatically increase the active earth pressure acting on the retaining walls, increasing the probability of instability. In this paper, an analytical solution to the Laplace differential governing equation is presented for seepage problems considering anisotropic permeability based on Fourier series expansion method. A good correlation is observed between this and the seepage forces along a planar surface generated via finite element analysis. The active earth pressure is calculated using Coulomb's earth pressure theory based on the calculated pore water pressures. The obtained solutions can be degenerated into Coulomb's formula when no seepage exists in the backfill. A parametric study on the influence of the degree of anisotropy in seepage flow on the distribution of active earth pressure behind the wall is conducted by varying ratios of permeability coefficients in the vertical and horizontal directions,showing that anisotropic seepage flow has a prominent impact on active earth pressure distribution. Other factors such as effective internal friction angle of soils and soil/wall friction conditions are also considered.