Читать книгу The Rheology Handbook - Thomas Mezger - Страница 26

Assumptions: fluid in a state-at-rest; the particles are almost suspended and therefore they are sinking very, very slowly in a steady-state process (laminar flow, at a Reynolds number Re ≤ 1; more about Re numbers: see Chapter 10.2.2.4b); spherical particles; the values of the weight force FG [N] and the flow resistance force FR [N] of a particle are approximately equal in size.

According to Stokes’ law (Georges G. Stokes, 1819 to 1903 [2.12]):

Equation 2.6

FG = Δm ⋅ g = FR = 3 ⋅ π ⋅ dp ⋅ η ⋅ v

with the mass difference Δm [kg] between a particle and the surrounding fluid, the gravitation constant g = 9.81 m/s2, the mean particle diameter dp [m], the shear viscosity of the dispersion fluid η [Pas], and the particles’ settling velocity v [m/s].

The following applies: Δm = Vp ⋅ Δρ, with the volume Vp [m3] of a particle, and the density difference Δρ [kg/m3] = (ρp - ρfl) between the particles and the dispersion fluid; particle density ρp [kg/m3] and fluid density ρfl [kg/m3].

The following applies for spheres: Vp = (π ⋅ dp 3) / 6; and therefore, for the settling velocity

Equation 2.7

Assumption for the shear rate: γ ̇ = v/h

with the thickness h of the boundary layer on a particle surface, which is sheared when in motion against the surrounding liquid (the shear rate occurs on both sides of the particle). This equation is valid only if there are neither interactions between the particles, nor between the particles and the surrounding dispersion fluid.

Assuming simply, that h = 0.1 ⋅ d, then: γ ̇ = (10 ⋅ v)/d