牛頓或卡羅流體中粒子間或與邊界之交互作用

The Interaction between Flocs or Solid Particles with Boundary Effect in Newtonian and Carreau Fluid

葉淑仁(Shu-Jen Yeh)

指導教授 ： 徐治平

國立臺灣大學/工學院/化學工程學研究所/博士(2008年)

https://doi.org/10.6342/NTU.2008.01022

摘要

本論文以有限元素法軟體FIDAP 7.6研究不同幾何系統下膠羽或實心球粒子們在牛頓或非牛頓流體中運動而彼此相互影響的現象。類似流動系統中分佈多顆粒子的現象在工業上相當常見，如流體化床；而計算一顆粒子流經一個流體時所受的拖曳力卻是個很典型的議題。實際上，在計算粒子所受的拖曳力時，有兩個很重要的因素應該被考慮，一個即是邊界的存在與否，如容器的牆壁、底部；另一個是鄰近粒子的影響，特別是當粒子濃度不可忽略時。所以本文研究的方向在於兩顆非均勻球形膠羽相互作用下，對其流動效應的影響，所得結果可應用在如與廢水處理相關的操作程序中。另外，粒子在沉降槽中做沉降運動時，經常會受到如沉降槽的槽壁、槽底等邊界的影響，故本文亦探討在邊界效應可能重要的情況下，兩顆非均勻球形膠羽或兩顆實心球粒子彼此相互作用，而對流動效應的影響。文中亦討論採用free surface cell model來模擬實心球粒子群在非牛頓流體中的群體效應。

在描述多孔體在流體中運動的行為時，由於膠羽實際的組成結構相當複雜，為簡化問題，我們假設整個膠羽粒子為一具有不同穿透係數的雙層多孔性物質，經由改變該兩層結構的穿透度模擬各類可能的膠羽結構。而其所需用到的方程式包括，描述流體在多孔體外運動的Navier-Stokes方程式與描述流體在多孔體中運動的Darcy-Brinkman model，並以數值方法求解該組高度非線性的主控方程式。

數值模擬的結果顯示，下列幾個因素對膠羽或實心球粒子的拖曳力有著重大的影響：膠羽內、外層結構的穿透度比與其半徑比、粒子與粒子間的距離、邊界存在的重要程度、粒子與邊界的幾何形狀、系統內流體的本質。我們發現不論是在膠羽或是實心球粒子系統中，當雷諾數小時，不論兩顆粒子間距離的遠近，其作用在兩顆粒子上的拖曳力是相等的；但當雷諾數增加時，由於流體流經leading particle時，會在其後方形成漩渦，所以作用在leading particle的拖曳力會大於rear particle所受到的。而在膠羽系統中，由於膠羽粒子的非均勻性，當兩顆膠羽在自由沉降時，在相同的雷諾數及體積平均穿透度下，結構愈不均勻的膠羽，所之受拖曳力愈大；且在雷諾數為40時，其拖曳係數對雷諾數作圖與Stokes’-law like correlation比較，會有著較大的偏離現象，而此偏差亦會隨著膠羽間彼此的影響變大而愈趨明顯。在邊界存在的情況下，如管壁或是沉降槽的底部…等，流體會被邊界所壓縮，且即使當膠羽粒子在體積平均穿透度相同的情況下(如：ko/ki=0.1和ko/ki=10)，流場仍然會受膠羽本身內部的結構所影響；與在無邊界存在時的結果比較，邊界存在的重要程度有助於拖曳係數對雷諾數作圖與Stokes-law like correlation之間的偏差程度縮小。至於在實心球粒子系統中，邊界的存在會加速流體的對流運動，使得粒子後方的流速逆轉並產生邊界層的分離，而此現象在當流體的shear-thinning性質明顯時愈趨於顯著。由於在實心球粒子系統中，邊界的幾何形狀、粒子與粒子間排列的方位、流體的性質，此些範圍討論的較為廣泛，所以在一些定性、定量的分析上較無一規則性的變化，如：邊界或粒子濃度對拖曳係數影響的程度、拖曳係數對雷諾數作圖與Stokes’-law like correlation之間的偏差程度，常會隨著兩顆粒子間的距離、流體的shear-thinning性質、雷諾數…等的變化而出現一局部的極小值。最後本論文也根據數值模擬的結果，針對某些系統進行迴歸分析，得出拖曳係數與各個參數間之經驗關係式。

關鍵字

拖曳力 ； 牛頓流體 ； 卡羅流體 ； 邊界效應 ； 膠羽

並列摘要

This article uses FIDAP 7.6 based on finite element method to make a research on the interaction between moving rigid spheres or flocs in Newtonian or non-Newtonian fluid under different geometric system. Such kind of particle distribution in flow system usually can be seen in many industries, such as fluidized bed. The evaluation of the drag acting on a particle as it translates in a fluid medium is of both fundamental and practical significance. In practice, two important factors usually should be considered in the calculation of the drag on a particle, namely, the presence of a boundary such as container wall or the bottom of a tank, and the influence of neighboring particles, especially when the concentration of particles is appreciable. Thus, in this article we mainly make a study on the floc about its behavior in a flow field when two non-uniformly structured flocs interacts each other. The results obtained can be applied in the operating process of waste water treatment. In practice, it is conducted in a space where the boundary effect can play a significant role. Therefore, the discussion of the influence on the particle on its behavior in a flow field due to the interaction between two non-uniform spherical flocs or two rigid particles when boundary effect might be important is also provided in this text. The free surface cell model is also applied in this article to simulate the dispersion effect of an assemblage of spherical particles in non-Newtonian fluid.
Due to the fact that floc formation involves various nonlinear, random processes, its structure is of complicated nature. Here, a two-layer model is adopted to simulate the behavior of a floc where various possible floc structures are simulated by varying the permeability of each layer when describing the behavior of a moving porous particle in the flow field. The governing equation of the flow field inside and outside a floc, which is highly nonlinear and described by the Navier-Stokes equation coupled with Darcy-Brinkman model, is solved numerically for the case of low to medium Reynolds numbers, and the drag acting on the flocs are investigated.
The results obtained from numerical simulations show that the following factors has significant influences on the drag of flocs or solid particles: the ratios (inner radius/outer radius) and (permeability of outer layer/permeability of inner layer), the separation distance between two particles, how significant of the boundary effect is, the geometries of a particle and a boundary, and the nature of a fluid. We show that in both flocs or solid particles system, if Reynolds number is small, the drag on the leading particle is about the same as that on the rear particle no matter the distance between two particles is large or small. However, if Reynolds number is increasing to sufficiently large, because wakes are formed in the rear region of the leading particle, the drag on it is greater than that on the rear particle. The non-uniform of flocs in a floc system is significant. When two flocs are free settling under a fixed Reynolds number and volume-average permeability, the more non-uniform the floc structure is, the more important its influence on the drag. When Reynolds number is 40, the deviation of the drag coefficient-Reynolds number curve from a Stokes’-law-like relation is large and become more appreciable as the increase of interaction between particles. The presence of a boundary, such as the wall of a tube or the bottom of a settling tank, makes the fluid compressed and the flow field is affected by the structure inside the floc even if the mean permeability is the same (ex: ko/ki=0.1 and ko/ki=10). This implies that the prediction of the flow field based on the mean property of a floc is unsatisfactory, and knowledge about its detailed structure is necessary. Comparing with the condition that there no boundary exists, the presence of the boundary has the effect of reducing that deviation. As in solid particle system, the presence of the boundary has the effect of enhancing the convective motion in the rear part of a sphere, thereby forming wakes and a reverse flow field, and this phenomenon is enhanced by the shear-thinning nature of a fluid. The discussion is wide-ranged in solid particle system, including the geometries of the boundary, the arrangement between particles, and the nature of a fluid, therefore that the changes or behaviors in qualitative and quantitative analysis are not so regular. Such as that the extent boundary or particle concentration influences drag coefficient and the deviation of the drag coefficient-Reynolds number curve from a Stokes’-law-like relation has a local minimum as the change of the separation distance between two particles, the shear-thinning nature of fluid, Reynolds number…etc. Finally that in this article, a regression analysis is conducted based on the simulation data gathered in our study and the empirical relations among drag coefficient and every parameters is obtained.

並列關鍵字

drag force ； Newtonian fluid ； Carreau fluid ； boundary effect ； floc

參考文獻

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