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4.5: The Shape Factor

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  • In the range of particle Reynolds numbers from roughly unity to about 100, which is the range of interest here, a particle orients itself during settling so as to maximize drag. Generally this means that an oblate or lenticular particle, i.e. a shape with one dimension smaller than the other two, will settle with its maximum area horizontal. The drag of fluid on the particle then depends most critically on this area. This is also the area seen if the particle lies in a stable position on a flat surface. Therefore, for estimation of drag, the non-spherical particle is characterized by the ‘area equivalent diameter’, i.e. the diameter of the sphere with the same projected area. For particles whose sizes are determined by sieving rather than microscopic analysis, the diameter is slightly smaller than the mesh size. However, unless the particles are needle shaped, the difference between the equivalent diameter and the screen opening is relatively small, generally less than 20%.

    Although equation (4.2-5) contains a shape factor, basically all the equations in this chapter are derived for spheres. The shape factor ψ in equation (4.2-5) is one way of introducing the effect of the shape of particles on the terminal settling velocity. In fact equation (4.2-5) uses a shape factor based on the weight ratio between a real sand particle and a sphere with the same diameter. Another way is introducing a factor ξ according to:

    \[\ \xi=\frac{\mathrm{v}_{\mathrm{t}}}{\mathrm{v}_{\mathrm{ts}}}\]

    Where ξ equals the ratio of the terminal settling velocity of a non-spherical particle vt and the terminal velocity vts of a spherical particle with the same diameter. The shape of the particle can be described by the volumetric shape factor which is defined as the ratio of the volume of a particle and a cube with sides equal to the particle diameter so that K=0.524 for a sphere:

    \[\ \mathrm{K}=\frac{\text { volume of particle }}{\mathrm{d}^{3}}\]

    The shape factor ξ is a function of the volumetric form factor and the dimensionless particle diameter D* according to equation (4.4-21).

    \[\ \log (\xi)=-0.55+\mathrm{K}-0.0015 \cdot \mathrm{K}^{2}+0.03 \cdot 1000^{\mathrm{K}-0.524}+\frac{-0.045+0.05 \cdot \mathrm{K}^{-0.6}-0.0287 \cdot 55000^{\mathrm{K}-0.524}}{\cosh \left(2.55 \cdot\left(\log \left(\mathrm{D}^{*}\right)-1.114\right)\right.}\]

    This equation takes a simpler form for sand shaped particles with K=0.26:

    \[\ \log (\xi)=-0.3073+\frac{0.0656}{\cosh \left(2.55 \cdot \log \left(D^{*}\right)-1.114\right)}\]

    A value of K=0.26 for sand grains would give a volume ratio of 0.26/0.524=0.496 and thus a factor ψ=0.496 in equation (4.2-5), while often a factor ψ=0.7 is used.

    Figure 4.5-1 shows the shape factor ξ as a function of the dimensionless particle diameter D*, according to equation (4.5-3).

    Figure 4.4-3 also shows the terminal settling velocity according to the methods of Huisman (1973-1995) and Grace (1986) using the shape factor according to equation (4.5-4). It can be seen clearly that both methods give the same results. One can see that the choice of the shape factor strongly determines the outcome of the terminal settling velocity.

    Figure 4.5-1: The shape factor ξ as a function of the dimensionless particle diameter D*.

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