Determining the velocity of particles settling through fluids is of great importance of many areas of engineering

Question:

Determining the velocity of particles settling through fluids is of great importance of many areas of engineering and science. Such calculations depend on the flow regime as represented by the dimensionless Reynolds number,

Re =pdv/μ (P8.48.1)

where ρ = the fluid’s density (kg/m³), d = the particle diameter (m), y = the particle’s settling velocity (m/s), and μ = the fluid’s dynamic viscosity (N s/m²). Under laminar conditions (Re, 0.1), the settling velocity of a spherical particle can be computed with the following formula based on Stokes law,

where g = the gravitational constant (5 9.81 m/s²), and p_s = the particle’s density (kg/m³). For turbulent conditions (i.e., higher Reynolds numbers), an alternative approach can be used based on the following formula:

where C_D = the drag coefficient, which depends on the Reynolds number as in

(a) Combine Eqs. (P8.48.2), (P8.48.3), and (P8.48.4) to express the determination of y as a roots of equations problem. That is, express the combined formula in the format f(y) = 0.

(b) Use the modified secant method with δ = 10^–3 and ε_s = 0.05% to determine v for a spherical iron particle settling in water, where d = 200 μm, ρ = 1 g/cm³, ρ_s = 7.874 g/cm³, and μ = 0.014 g/(cm∙s). Employ Eq. (P8.48.2) to generate your initial guess.

(c) Based on the result of (b), compute the Reynolds number and the drag coefficient, and use the latter to confirm that the flow regime is not laminar.

(d) Develop a fixed-point iteration solution for the conditions outlined in (b).

(e) Use a graphical approach to illustrate that the formulation developed in (d) will converge for any positive guess.