The Rheology Handbook. Thomas Mezger

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       Figure 2.3: Velocity distribution and shear rate in the shear gap of the Two-Plates model

      Both γ ̇ and v provide information about the velocity of a flowing fluid. The advantage of selecting the shear rate is that it shows a constant value throughout the whole shear gap. Therefore, the shear rate is independent of the position of any flowing layer in the shear gap. Of course, this applies only if the shear conditions are met as mentioned in the beginning of Chapter 2.2. However, this does not apply to the velocity v which decreases from the maximum value vmax on the upper, movable plate to the minimum value vmin = 0 on the lower, immovable plate. Therefore, when testing pure liquids, sometimes as a synonym for shear rate the term velocity gradient is used (e. g. in ASTM D4092).

      b) Calculation of shear rates occurring in technical processes

      The shear rate values which are given below are calculated using the mentioned formulas and should only be seen as rough estimations. The main aim of these calculations is to get merely an idea of the dimension of the relevant shear rate range.

      1) Coating processes: painting, brushing, rolling or blade-coating

       γ ̇ = v/h, with the coating velocity v [m/s] and the wet layer thickness h [m]

      Examples

      1a) Painting with a brush:

      With v = 0.1 m/s and h = 100 µm = 0.1 mm = 10-4 m; result: γ ̇ = 1000 s-1

      1b) Buttering bread:

      With v = 0.1 m/s and h = 1 mm = 10-3 m; result: γ ̇ = 100 s-1

      1c) Applying emulsion paint with a roller

      With v = 0.2 m/s (or 5 s per m), and h = 100 µm = 0.1 mm = 10-4 m; result: γ ̇ = 2000 s-1

      1d) Blade-coating of adhesive dispersions (e. g. for pressure-sensitive adhesives PSA):

      With the application rate AR (i. e. mass per coating area) m/A [g/m2]; for the coating volume V [m3] applies, with the mass m [kg] and the density ρ [1 g/cm3 = 1000 kg/m3]: V = m/ρ

      Calculation: h = V/A = (m/ρ)/A = (m/A)/ρ = AR/ρ; with AR = 1 g/m2 = 10-3 kg/m2 holds:

      h =10-6 m = 1 µm; and then: γ ̇ = v/h. See Table 2.2 for shear rates occurring in various kinds of blade-coating processes [2.4] [2.5].

      2) Flow in pipelines, tubes and capillaries

      Assumptions: horizontal pipe, steady-state and laminar flow conditions (for information on laminar and turbulent flow see Chapter 3.3.3), ideal-viscous flow, incompressible liquid. According to the Hagen/Poiseuille relation , the following holds for the maximum shear stress τw and the maximum shear rate γ ̇ w in a pipeline (index w for “at the wall”):

      Equation 2.4

      τw = (R ⋅ Δp) / (2 ⋅ L)

      Equation 2.5

       γ ̇ w = (4 ⋅ V ̇ ) / (π ⋅ R3)

      With the pipe radius R [m]; the pressure difference Δp [Pa] between inlet and outlet of the pipe or along the length L [m] of the measuring section, respectively (Δp must be compensated by the pump pressure); and the volume flow rate V ̇ [m3/s]. This relation was named in honor to Gotthilf H. L. Hagen (1797 to 1848) СКАЧАТЬ