(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
Examples
3a) Sedimentation of sand particles in water
With dp = 10 µm = 10-5 m, η = 1mPas = 10-3 Pas, and ρp = 2.5 g/cm3 = 2500 kg/m3 (e. g. quartz silica sand), and ρfl = 1 g/cm3 = 1000 kg/m3 (pure water); results: v = approx. 8.2 ⋅ 10-5 m/s
Such a particle is sinking a maximum path of approx. 30 cm in 1 h (or approx. 7 m per day).
With h = 1 µm results: γ ̇ = v/h = approx. 80 s-1
3b) Sedimentation of sand particles in water containing a thickener
With dp = 1 µm = 10-6 m, and η = 100 mPas (e. g. water containing a thickener, measured at
γ ̇ = 0.01 s-1), and with the same values for ρp and ρfl as above in Example (3a), results: v = approx. 8.2 ⋅ 10-9 m/s (or v = 0.7 mm per day). With h = 0.1 µm results: γ ̇ = 0.08 s-1 approximately.
Note 1: Calculation of a too high settling velocity if interactions are ignored
Stokes’ sedimentation formula only considers a single particle sinking, undisturbed on a straight path. Therefore, relatively high shear rate values are calculated. These values do not mirror the real behavior of most dispersions, since usually interactions are occurring. The layer thickness h is hardly determinable. We know from colloid science: It depends on the strength of the ionic charge on the particle surface, and on the ionic concentration of the dispersion fluid (interaction potential, zeta-potential) [2.28] [2.29]. Due to ionic adsorption, a diffuse double layer of ions can be found on the particle surface. For this reason, in reality the result is usually a considerably lower settling velocity. Therefore, and since the shear rate within the sheared layer is not constant: It is difficult to estimate the corresponding shear rate values occurring with sedimentation processes.
Note 2: Particle size of colloid dispersions, and nano-particles
In literature, as medium diameters of colloid particles are mentioned different specifications: between 10-9 m and 10-6 m (or 1 nm to 1 µm) [2.14] [2.25], or between 10-9 m and 10-7 m (or 1 nm to 100 nm) [2.13], or between 10-8 m and 10-6 m (or 10 nm to 1 µm) [2.26]. In ISO 80004-1 of 2015 is stated: Nano-scaled particles are in the range of approximately 1 nm to 100 nm [2.27]. Due to Brownian motion, the nano-particles usually are remaining in a suspended state and do not tend to sedimentation. Above all, the limiting value of the settling particle size depends on the density difference of particles and dispersing fluid.
2.2.3Viscosity
For all flowing fluids, the molecules are showing relative motion between one another, and this process is always combined with internal frictional forces. Therefore, for all fluids in motion, a certain flow resistance occurs which may be determined in terms of the viscosity. All materials which clearly show flow behavior are referred to as fluids (thus: liquids and gases ).
a) Shear viscosity
For ideal-viscous fluids measured at a constant temperature, the value of the ratio of shear stress τ and corresponding shear rate γ ̇ is a material constant. Definition of the shear viscosity, in most cases just called “viscosity“:
Equation 2.8
η = τ/ γ ̇
η (eta, pronounced: etah or atah), the unit of the shear viscosity is [Pas], (pascal-seconds).
The following holds: 1Pas = 1N ⋅ s/m2 = 1 kg/s ⋅ m
For low-viscosity liquids, the following unit is usually used:
1 mPas (milli-pascal-seconds) = 10-3 Pas
Sometimes, for highly viscous samples the following units are used:
1 kPas (kilo-pascal-seconds) = 1000 Pas = 103 Pas, or even
1 MPas (mega-pascal-seconds) = 1000 kPas = 1,000,000 Pas = 106 Pas
A previously used unit was [P], (“poise”; at best pronounced in French); and: 1 P = 100 cP; however, this is not an SI unit [2.15]. This unit was named in honor to the doctor and physicist Jean L. M. Poiseuille (1799 to 1869) [2.7].
The following holds: 1 cP (centi-poise) = 1 mPas, and 1 P = 0.1 Pas = 100 mPas
Sometimes, the term dynamic viscosity is used for η (as in DIN 1342-1). However, some people use the same term to describe either the complex viscosity determined by oscillatory tests, or to mean just the real part of the complex viscosity (the two terms are explained in Chapter 8.2.4b). To avoid confusion and in agreement with the majority of current international authors, here, the terms viscosity or shear viscosity will be used for η. Table 2.3 lists viscosity values of various materials.
The inverse value of viscosity is referred to as fluidity Φ (phi, pronounced: fee or fi) [2.17]. However today, this parameter is rarely used. The following holds:
Equation 2.9
Φ = 1/η with the unit [1/Pas] = [Pas-1]
| Table 2.3: Viscosity values, at T = +20 °C when without further specification; own data and from [2.2] [2.3] [2.16] | |
| Material | Viscosity η [mPas] |
| gases/air | 0.01 to 0.02 / 0.018 |
| pentane/acetone/gasoline, petrol
| |