以離散元素模擬探討摩擦係數對顆粒流動行為之影響

Abstract

微觀顆粒的互動行為與巨觀顆粒流的物理現象之間的關係一直是重要的研究議題。近年來，由於數值模擬愈趨成熟，許多過去單憑實驗很難觀察到的微觀顆粒行為，例如：顆粒之間的接觸力與相對速度，都可以透過數值模擬得到合理的結果，進而建立和實驗觀察到的巨觀現象之間的關係。在顆粒流的研究中，離散元素法(discrete element method, DEM)是受到許多學者採用的數值模擬方法，以其能夠進行大量且快速的模擬並得到和實驗相符的結果。 慣性數I是一個能夠捕捉顆粒流特徵的無因次參數，根據I值的大小，可以得知顆粒流的行為比較接近固體還是流體。近年來有學者拿I和有效摩擦係數μ之間的關係來推導顆粒流的組成律，使得I和μ成為研究顆粒流非常重要的巨觀參數。透過DEM的模擬，有學者指出顆粒流的巨觀有效摩擦係數μ與微觀摩擦係數f的關係不大。然而，亦有學者發現當f≤0.4，μ值隨著f呈線性成長。為了釐清f和μ之間的關係，本研究透過DEM進行剪力槽(shear cell)實驗的模擬，發現在相近的慣性數I之下，剪力槽內顆粒流的μ隨著f的增加而增加。這是因為在剪力槽內，f對壁體承受的正向力的影響相對不大。同時，f的增加會提高發生庫倫滑動的門檻，讓顆粒和顆粒以及顆粒和壁體之間產生較大的切向力，造成μ值的上升。此外，本研究發現由於和邊界接觸的顆粒會發生「鐘擺效應」機制，顆粒流和邊界之間的接觸切向力方向不會一致，造成穩態顆粒流的μ明顯低於f。另一方面，當剪力槽的轉速夠快，暫態顆粒流和邊界的切向接觸力有機會完全進入庫倫摩擦。此時所有和邊界接觸的顆粒受到方向一致的摩擦力作用，本研究稱之為「完全同步」狀態。「完全同步」下的顆粒流的μ值幾乎和f一樣。 微觀摩擦係數f和旋轉鼓內速度剖面曲線的關係也是本研究探討的議題。當旋轉鼓內的顆粒流達到穩態，該顆粒系統會因為具有較高的f而形成較大的動態安息角β，具有較高的位能。然而，旋轉鼓內速度剖面曲線並沒有因為顆粒系統的f不同而有顯著的變化，暗示具有不同f的穩態顆粒系統之間動能的差異遠不及彼此位能的差異，也代表微觀摩擦係數f會影響顆粒系統內部顆粒所作的功。本研究透過「功能原理」的應用以及使用DEM進行微觀能量的計算，發現不同f的顆粒系統從靜止到「動能初峰期」這段時間系統內接觸力所做的功Π*會決定顆粒系統在穩態期間的位能。換句話說，不同f的穩態顆粒系統彼此之間的位能差異，主要來自於彼此Π*的差值。本研究進一步分析每種接觸力所作的功隨著f的變化，發現從靜止到「動能初峰期」這段時間內具有較大f的顆粒系統並沒有消散比較多的能量，而是透過切向彈簧力獲得較多的能量。

The relation between the interactions of microscopic particles and phenomena of macroscopic granular flow have attracted great interests recently. As the numerical simulation techniques improve, many quantities of microscopic particles that are difficult to be observed or measured by experiments can be ready to computed. For example, the contact forces and relative velocities between particles can be calculated by simulation schemes. Thus, the relations between microscopic behaviors and macroscopic phenomena can be found. For granular flows, one of the most popular numerical simulation schemes is the discrete element method (DEM), because it can process large-scale particle simulations and produce numerical results in a good agreement with experimental measurements. The inertial number, I, is a non-dimensional parameter to quantify the flowing regime of granular flows. Recently, constitutive laws of granular flows have been derived by using the relations between I and the macroscopic effective friction coefficient, μ. Through DEM simulation, some studies suggested that the macroscopic effective friction coefficient was independent of the microscopic friction coefficient, f. However, there were studies claimed that μ was linearly dependent on f when f≤0.4. In order to improve the understanding of the relation between μ and f, shear cell simulations are performed in this research. The macroscopic effective friction coefficients of the granular flows in the shear cell are found to monotonically grow with microscopic friction coefficients under similar values of I, because the influence of f on normal forces is much less than tangential forces on the cell wall. At the same time, the increasing f raises the criterion of Coulomb friction and allows larger tangential forces between particles and cell wall, thus resulting in the increasing of μ. Besides, because the pendulum effect occurs on the particles on the cell wall, the directions of tangential forces between particles and cell wall are not identical, thus resulting in the values of μ from steady state granular flows much less than f. On the other hand, when the rotating speeds of the shear cell are fast enough, all tangential forces between the particles and cell wall of transient granular flows reach the criterion of Coulomb friction. At this moment, all particles on the cell are driven by the friction forces with the same direction, named complete synchronization herein. The granular flows under the complete synchronization state have the same values of μ and f. The relation between the microscopic friction coefficients and velocity profiles of particle systems in rotating drums is the other research issue in this thesis. When the particle systems, half-filled the drum, reach a steady state, the ones with higher f will form higher dynamic repose angles, thus resulting in higher potential energies. However, the velocity profiles of the particle systems in the rotating drum are not obviously influenced by f. These similar velocity profiles suggest that the particle systems with different f have similar kinetic energy. Although the particle systems with higher f have higher potential energies, the kinetic energies are almost identical, which indicates f has significant effects on the work of internal forces between particles. In this research, through the application of principle of work and energy and the calculation of microscopic energy using the DEM simulation, the potential energies of steady state particle systems with different f are found determined by the work, Π*, done by contact forces during the first kinetic peak period. In other words, the difference of potential energies of particle systems with different f comes from the difference of their Π*. Furthermore, to analyze the components of Π* with different f, the particle systems with higher f are found not dissipating more energy, but obtaining more energy from tangential contact force.