1.School of Physics and Electronics, Central South University, Changsha 410083, China 2.Powder Metallurgy Research Institute, Central South University, Changsha 410083, China
Abstract:In previous work [Acta Phys. Sin. 60 054501 (2011)], we found that for inclined Granular Orifice Flow (GOF) in air, regardless of the orifice size, the flow rate Q had a good linear relationship with the cosine of the inclination $\cos \theta $, i.e. $\dfrac{Q}{{{Q_0}}} = 1 - \dfrac{{\cos \theta - 1}}{{\cos {\theta _{\rm c}} - 1}}$, where Q0 is flow rate at $\theta ={0^ \circ }$, and ${\theta _{\rm c}}$ is the critical angle of flow ceasing obtained by linear extrapolation. Moreover, ${\theta _{\rm c}}$ increased linearly with ratio between grain and orifice diameter d/D, and at the limit of d/D going to zero (that is, D going to infinity), the angle of repose of the sample ${\theta _{\rm r}}~( = 180^ \circ - \theta _{\rm c\infty})$ was obtained. Since the flow of GOF is very stable, we believe that the linear extrapolation of the above-mentioned inclined GOF provides a novel method for accurately measuring the angle of repose of granular materials. This method has been proved to be effective in a wider orifice size range by another work [Acta Phys. Sin. 65 084502 (2016)]; and three angles, namely the repose angle measured by GOF, the free accumulation angle of a sandpile and the internal friction angle of the granular material measured by Coulomb yielding, are confirmed to be consistent. In this work, we extend this method to underwater, measuring the mass flow rate of a granular sample (glass beads) which completely immersed in water and driven by gravity, discharged from an inclined orifice for various inclination angles and orifice diameters. It is found that similar to the case in air, regardless of the orifice size, the flow rate increase linearly with the cosine of the inclination; the critical angle of flow ceasing increases linearly with ratio between grain and orifice diameter; at the limit of infinite orifice, this critical angle is consistent with the repose angle of the underwater sample within the experimental error range. In addition, all measurements can be well fitted by using the Beverloo formula $Q = {C_0}\rho {g^{1/2}}{(D - kd)^{5/2}}$, where the parameters C0 and k are only related to the cosine of the inclination, and are linear and inversely squared, respectively. Compared with the results of GOF in air reported by previous work, it is found that the difference mainly comes from the influence of buoyancy and fluid drag forces on the parameter C0. These results show that both the method of measuring angle of repose with the inclined GOF and the Beverloo formula have certain universality. The behavior of GOF is qualitatively the same whether the interstitial fluid is water or air. Keywords:granular matter/ inclined orifice flow/ angle of repose/ Beverloo formula
2.实验装置与实验结果如图1(a)所示, 实验的主要结构是一个部分浸泡在水中的铝合金方形料仓, 料仓横截面为100 mm × 100 mm, 高700 mm, 壁厚3 mm. 颗粒样品采用直径$d = 0.9 \pm 0.1{\rm{ }}\;{\rm{mm}}$的球形玻璃珠, 在实验过程中颗粒样品始终位于水面以下, 处于饱和水状态. 悬挂着料仓的刚性悬臂由两个调节过水平的电子秤支撑(精度为0.5 g, 量程为30 kg, 采样频率为10 Hz). 电子秤固定在刚性支架上, 示数之和就是料仓的总质量$M(t)$(包含了不随时间变化的仓壁自重和随时间变化的颗粒质量). 实验中随着颗粒的流出, 电子秤示数$M(t)$随时间减少, 减少率即为GOF的流量Q. 图 1 (a)实验装置示意图; (b)料仓透水侧壁照片; (c)倾角小于45度时采用的实验装置; (d)倾角大于45度时采用的实验装置; (e)楔形孔洞示意图 Figure1. (a) Schematic of the setup; (b) photograph of the permeable side wall of the silo; (c) the experimental devices used when the inclination is less than 45 degrees; (d) the experimental devices used when the inclination is greater than 45 degrees; (e) schematic of the wedge-shaped orifice D.
当颗粒从料仓中流出时, 料仓内部减少的颗粒体积需要及时由水填补, 以维持料仓内外水面高度不变和样品的水饱和状态, 从而防止内外的液压差对颗粒孔洞流量的影响. 为此料仓的四个侧壁均镶嵌了密排冲孔(孔径为0.8 mm)的不锈钢板(图1(b))用以透水, 以保证实验过程中料仓内外液面保持高度一致(也保证了料仓壁在实验过程中受到水的浮力不变). 与文献[16]类似, 实验测量了不同倾角$\theta $下, 7个孔径$D$=4, 6, 8, 10, 14, 18, 20 mm的$M(t)$结果. 倾角$\theta $定义为圆孔所在平面与外水平面的夹角, 如图1(c)和图1(d)所示. 倾角大小由精度为0.1度的倾角仪(bevel box)测出. 利用底部开孔的装置(图1(c))可测量$\theta \in [{0^ \circ }, {45^ \circ }]$的颗粒流量. 对于$\theta > {45^ \circ }$的情形, 需要采用图1(d)所示的料仓侧壁开孔装置. 在距离底部约100 mm的高度开一个100 mm × 100 mm的方孔, 用以嵌入圆孔$D$所在的硬铝板. 为尽量减小板厚度对流量的影响, 孔洞外侧加工成了如图1(e)所示的楔形. 实验中发现, 如果将颗粒直接从空气置入水中的料仓, 颗粒表面会裹挟大量空气泡, 这些气泡在流动时会逐渐释出, 对孔洞流量带来显著影响. 在孔洞较小时气泡甚至会直接阻塞颗粒流. 因此在开始实验之前, 需要提前将颗粒放入水桶中, 让其在水中反复多次流动, 使气泡充分释出直至肉眼不可见. 之后再将去气泡后的颗粒通过开口的侧壁在液面以下加入料仓, 并且在加料的过程中始终保证颗粒不再暴露到空气中. 通过去气泡处理后的颗粒流量将十分稳定和可重复. 实验开始前先将孔$D$塞住, 在料仓中加入去气泡处理过的颗粒直至接近水面, 然后开放孔$D$, 同时采集电子秤随时间变化的$M(t)$. 典型的$M(t)$数据如图2所示. 由图2可以看出, 在初期GOF有一段短暂的不稳定流动期(包括开放孔D带来水面波动的影响), 但之后$M(t)$呈现出良好的直线关系, 其斜率的负值就是GOF的流量$Q$(由于测量的是料仓中颗粒的减少量, 故$M(t)$斜率为负). 这个流量在很长一段时间内是稳定不变的, 如图2插图所示, 40—80 s和80—120 s计算出来的流量$Q$均为6.53 g/s. 在流动的尾声(140 s左右), $M(t)$曲线只有非常微弱(小于2%)的变陡趋势. 没有明显地观察到文献[19]中提到的在料仓快流完时流量变大的现象. 这个分歧的原因目前不详. 但快流完的情况不是关注的重点, 本文仅研究中间段的稳定流量$Q$随$D$和$\theta $的变化关系. 图 2$D = 14$ mm, $\theta = 90^\circ $ 时典型的$M(t)$数据; 左下和右上插图分别为从主图中摘出的40?80 s及80?120 s的$M(t)$数据, 均呈良好的线性关系. 由这两段数据计算得到的流量$Q$没有差别, 均为6.53 g/s, 表明流量非常稳定 Figure2. Typical data of M(t) at D = 14 mm, $\theta = 90^\circ $;the lower left and upper right insets are the data of 40?80 s and 80?120 s extracted from the main graph, both of which show good linearity. Both flow rates$Q$ calculated from these two insets are 6.53 g/s, indicating that the flow is very stable.
对于每一给定的$D$和$\theta $重复三次实验, 平均值记为对应的流量$Q(D, \theta )$, 结果见图3(a)所示. 虽然水中操作的扰动比空气中的实验略大, 但在去除气泡后水中的GOF流量仍然非常稳定, 实验重复性好, 三次测量的偏差均小于数据点图标的大小, 因此图3a的实验数据点没有标出误差棒. 图 3 (a)不同孔径D下流量Q随倾角余弦$\cos \theta $ 的变化, 实线为直线拟合; (b)用水平($\theta = {0^\circ }$)流量${Q_0}$ 归一化的流量$Q/{Q_0}$ 随倾角余弦$\cos \theta $ 的变化关系, 实线为公式(3)的拟合结果; (c)临界流量休止角${\theta _{\rm{c}}}$随粒径-孔径比d/D的变化关系, 实线和${\theta _0}$为直线拟合结果 Figure3. (a) The variation of flow rateQ with the inclination cosine $\cos \theta $ at different orifices D, where the solid line is a linear fit; (b) variation of the normalized flow rate $Q/{Q_0}$ with $\cos \theta $, where ${Q_0}$ is the rate at $\theta = {0^ \circ }$, and the solid line is the fitted result of equation (3); (c) the relationship between the critical angle of flow ceasing ${\theta _{\rm{c}}}$ and the ratio d/D, where the solid line and ${\theta _0}$ are results of linear fitting.
4.Beverloo公式图3的实验数据可以用公式(1)和(2)很好地拟合. 如图4(a)所示, 对所有不同的倾角$\theta $, 流量Q随D的变化关系都能很好地满足5/2次幂, 实验结果并不支持文献[18,19]提到的Q与$ {D}^{2} $成比例的结论. 这还可以在图4(a)的插图中, 从${Q^{2/5}}$和D的直线关系中得到验证. 图 4 (a)用Beverloo公式(1)和(2)拟合图3数据的结果, 插图为不同倾角时${Q^{2/5}}$ 随D的变化关系, 实线为线性拟合; (b), (c)Beverloo参数${C_0}$和$k$随$\cos \theta $ 的变化关系, 及其用公式(2)的拟合情况. 图(c)中的插图是${k^{ - 2}}$随$\cos \theta $的变化关系 Figure4. (a) Results of fitting the data in Figure 3 using the Beverloo formula (1) and (2), the inset is the change of ${Q^{2/5}}$ with D at different inclination, and the solid line is a linear fit; (b) and (c) variations of the parameters ${C_0}$ and $k$ with $\cos \theta $, and solid lines are results of fits by using equation (2). The inset in (c) is the change of ${k^{ - 2}}$ with $\cos \theta $.
由于幂次2/5和1/2的差别不大, 如果尝试拟合${Q^{1/2}}$与D的直线关系, 结果也显得不错(这可能也是文献[18,19]认为其合理的原因). 尽管两种幂次公式对本文数据的拟合相关系数(Adj. R-Square)均大于0.99(相关系数越接近1代表拟合程度越好), 2/5次幂的公式在拟合度上仍然要略高一点. 因此更倾向支持Beverloo公式(1)对于各种倾角的水中GOF均成立, 其中参数${C_0}$和$k$随倾角余弦$\cos \theta $的变化关系满足公式(2)(图4(b)和图4(c)). 虽然公式(2)对于水中和空气中的GOF都有效, 但在同样条件下水中的GOF流量明显小于空气中的流量(图5(c)—(f)). 这个流量差别, 主要由参数${C_0}$来描述(图5(a)). 参数$k$随$\cos \theta $的变化情况对水和空气不敏感(图5(b)). 另外从图5(c)—(f)的插图也可以看到, $k$对流量的影响也基本与水还是空气无关, 只在较大倾角时才出现一点差别. 在Beverloo公式中, $k$与孔径变小导致的流量阻塞停止(clogging)有关. 同样倾角下, 水中GOF与空气中GOF有着几乎同样的$k$, 显示这个阻塞机制与间隙流体的种类关系不大. 图 5 (a)和(b)是水中(实心方点)和空气中(空心圆点)GOF的Beverloo参数${C_0}$和k随$\cos \theta $的变化; (c)?(f)分别为$\theta = {0^ \circ }, {60^ \circ }, {90^ \circ }, {120^ \circ }$时, 水中和空气中GOF流量Q随D的变化, 插图是$Q/{C_0}$随D的变化. 空气中的实验数据来自文献[16] Figure5. (a) and (b): Beverloo parameters ${C_0}$ and $k$ of GOF in water (solid squares) and in air (hollow circles) as a function of $\cos \theta $; (c)?(f): the changes of GOF flow rate Q with D in water and in air when $\theta = {0^ \circ }, {60^ \circ }, {90^ \circ }, {120^ \circ }$, respectively, and the inset is the change of $Q/{C_0}$ with D. The experimental data in air comes from ref. [16].
从图6(a)可以看到, 水和空气的Beverloo系数${C_0}$的比值基本上是一个与倾角无关的常数0.37. 另外流量比值与0.37的偏差会随着孔径D的减小和倾角的增加而变大图6(b). 显然这个偏差反映的是Beverloo公式中系数$k$的效果. 值得指出的是, 间隙流体对颗粒的作用有浮力和粘滞拖曳力. 前者的效果可通过将重力加速度乘以密度变化比$\left( {\rho - {\rho _{\text{流体}}}} \right)/\rho $来描述($\rho = {\rho _{\text{颗粒}}}$是颗粒材料体密度). 对Beverloo公式(1)中的$g$做此处理, 得出水中与空气中GOF的${C_0}$比值为0.78(忽略空气密度), 明显大于图6(b)的0.37. 这意味着水的拖曳力(即粘滞力)对GOF流量有显著影响. 图 6 水和文献[16]空气的(a) Beverloo系数${C_0}$和(b) GOF流量$Q$的比值 Figure6. Ratio of (a) Beverloo coefficient ${C_0}$ and (b) GOF flow rate Q in water and in air (from Ref. [16])