Small particles moving through fluids by gravity

Small particles moving through fluids by gravity 

Let us consider a spherical particle that is surround by a fluid phase with a different density. The particle will move downwards through the action of gravity, and its motion will be checked by the drag force extended by the fluid. The equation of motion for the particle can be written as follows (with p = fluid density, = solid density and = drag coefficient) 

The drag coefficient relates the drag force to the kinetic energy of the fluid that is moved away by the particle. It is a constant when the fluid flow is turbulent. For spheres, for the range 10 Re 3.10, this constant is 0.43. However, when the particle starts to move, the flow is always laminar, and then the drag coefficient has to be replaced by 24/Re, so that the last equation becomes 

When the particles are relatively small, the fluid flow around the particles remains laminar, and after a while the velocity of the particles approaches a constant value. This terminal velocity, which is theoretically reached after infinite time, is found by taking dvidt = 0  

The mass transfer around a spherical particle moving with velocity v through fluid can be estimated from the following (Ranz and Marshall, 1952)  

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