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首页> 外文期刊>Zeitschrift fur Angewandte Mathematik und Mechanik >The motion of particles and sediment on an inclined plate
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The motion of particles and sediment on an inclined plate

机译：斜板上颗粒和沉积物的运动

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An important task in separation technology is separation of the smallest biological factors - proteins, viruses, antibodies, vaccines, etc. These particles are so small that Brownian motion may affect the sedimentation process. In particular, the ability to transport and remove sedimented particles along the walls of a centrifuge or a gravity settling device will crucially depend on the degree of packing of the particles in the sludge layer. Gradient diffusion of particles due to Brownian motion is in this respect to an advantage since it acts as to counteract the formation of high particle-density layers. In addition, if the sediment layer is not stationary but flows along the wall, yet another diffusion mechanism is present due to the shearing motion of the suspending fluid. This shear induced resuspension of particles was first observed by Galada-Maria (1979) and more recently studied by Leighton & Acrivos (1986, 1987 a, b). As described by these authors collisions between particles, as they are advected by the fluid, results in a random walk and drift of particles perpendicular to the plane of shear. A curve fit to the measured data of the effective diffusion coefficient of spherical particles given by Leighton & Acrivos (1986) is D_(coll) = a~2 γ 1/3 α~2(1 + 0.5 e~(8.8α)) where a is the particle radius, γ the shear rate and α the particle volume fraction. Nir & Acrivos (1990) who considered gravitational sedimentation and sediment flow on an inclined plate used this formula to obtain self similar steady state solutions for the case of infinitely large Peclet numbers, Pe = γμa~3 / kT 1, when Brownian diffusion is absent. They also showed that these boundary layer solutions are possible only in a certain regime of inclination angles, φ, and volume fraction, α_0, far away from the plate. If the inclination angle was too small or the volume fraction was too small or too large, maximum packing of particles would appear at the wall and prevent continuous transport of sediment down the plate. In this paper gravitational sedimentation of a homogeneous suspension on an inclined plate is reconsidered accounting both for the effects of Brownian diffusion and shear induced resuspension of particles. The additional term in the particle conservation equation, due to Brownian diffusion, unconditionally allows steady state solutions of the sediment boundary layer for all nonzero values of φ and α_0. Generally, the results show the development of a non self-similar boundary layer where the gravity settling towards the wall is balanced by Brownian diffusion at the beginning of the plate and by shear induced migration further downstream. Self-similar solutions may be approached asymptotically far downstream depending on the values of φ and α_0. For non self similar cases effects of Brownian diffusion persist in the near wall part of the boundary layer with dense particle packing and low shear.
机译：分离技术中的一项重要任务是分离最小的生物因子-蛋白质，病毒，抗体，疫苗等。这些颗粒非常小，以至于布朗运动可能会影响沉降过程。特别地，沿着离心机或重力沉降装置的壁运输和去除沉降的颗粒的能力将关键地取决于颗粒在污泥层中的堆积程度。在这方面，由于布朗运动引起的颗粒的梯度扩散是一个优点，因为它起到抵消高颗粒密度层形成的作用。另外，如果沉积物层不是静止的而是沿壁流动的，则由于悬浮流体的剪切运动而存在另一种扩散机制。这种剪切引起的颗粒重悬首先由Galada-Maria（1979）观察到，最近由Leighton＆Acrivos（1986，1987 a，b）研究。正如这些作者所描述的那样，粒子之间的碰撞（由流体推动）导致垂直于剪切平面的粒子随机游走和漂移。由Leighton＆Acrivos（1986）给出的与球形颗粒有效扩散系数的测量数据拟合的曲线为D_（coll）= a〜2γ1/3α〜2（1 + 0.5 e〜（8.8α））其中a是粒子半径，γ是剪切速率，α是粒子体积分数。 Nir＆Acrivos（1990）考虑了倾斜板上的重力沉降和泥沙流，对于无限大的Peclet数，当Brown扩散时，Pe =γμa〜3 / kT 1的情况，使用该公式获得自相似的稳态解。缺席。他们还表明，只有在一定角度的倾斜角φ和体积分数α_0（远离板）的情况下，这些边界层解决方案才有可能。如果倾斜角太小或体积分数太小或太大，则壁上会出现最大的颗粒堆积，从而阻止了沉积物沿板向下连续输送。在本文中，考虑了布朗扩散和剪切引起的颗粒重悬的影响，重新考虑了在倾斜板上均质悬浮液的重力沉降。由于布朗扩散，粒子守恒方程中的附加项无条件地允许对于φ和α_0的所有非零值进行沉积物边界层的稳态解。通常，结果显示了非自相似边界层的发展，其中朝向壁的重力沉降通过板开始处的布朗扩散和剪切诱导的下游迁移来平衡。自相似解可根据φ和α_0的值渐近地向下游移动。对于非自相似情况，布朗扩散的影响持续存在于边界层的近壁部分，具有密集的颗粒堆积和低剪切力。

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