Drag

Overview

The drag force F_D is typically quantified by the drag coefficient C_D:

F_D = \frac{1}{2} \rho v^2 C_D A

where \rho is fluid density, v is relative velocity, and A is the reference area.

Drag on Spheres

The flow over a sphere is a classic problem in fluid mechanics. The drag coefficient behavior changes dramatically with Reynolds number:

  1. Stokes Flow (Re < 0.1): Viscous forces dominate. C_D = 24/Re.
  2. Transition (0.1 < Re < 1000): Wake formation begins.
  3. Newtonian Regime (1000 < Re < 2\cdot 10^5): C_D is approximately constant (\approx 0.44).
  4. Drag Crisis (Re \approx 3\cdot 10^5): Boundary layer becomes turbulent, separation is delayed, and C_D drops sharply.
Figure 1: Terminal velocity of particles in water: Settling velocity vs. diameter for particles of different specific gravities (SG). Sand (SG=2.65), Steel (SG=7.85), Gold (SG=19.3).

Python Libraries

These functions utilize the fluids library, which contains industry-standard correlations for drag and terminal velocity. The library provides validated models that cover wide ranges of Reynolds numbers, from the Stokes regime to the drag crisis, and accounts for non-spherical particle geometries.

Calculators

  • DRAG_SPHERE: The primary function for drag calculations. It calculates C_D using user-selected correlations (standard, Clift-Gauvin, Morrison, etc.).
  • V_TERMINAL: Solves the iterative force balance to find the steady-state settling velocity. Crucial for designing sedimentation tanks, clarifiers, and fluidized beds.

Non-Spherical Particles

Real-world particles are rarely perfect spheres. Shape factors (sphericity) modify the drag characteristics. Correlations by Haider and Levenspiel (CD_HAIDER_LEVENSPIEL) or Ganser (CD_GANSER) account for particle irregularity.

Native Excel Capabilities

Excel lacks built-in support for drag and terminal velocity calculations. Most engineers use simplified formulas like Stokes’ Law, which is only valid for extremely low Reynolds numbers (Re < 0.1). Outside this regime, the calculation requires: - Selecting appropriate empirical correlations - Performing iterative solutions to balance drag, buoyancy, and weight - Accounting for non-spherical shapes

Manually implementing these iterative solvers in Excel (via Goal Seek or complex formulas) is inefficient and prone to convergence issues. The Python functions provided here automate these iterations and use validated correlations, significantly reducing analysis time and improving reliability.

Tools

Tool Description
CD_ALMEDEIJ Calculate drag coefficient of a sphere using the Almedeij correlation.
CD_BARATI Calculate drag coefficient of a sphere using the Barati correlation.
CD_BARATI_HIGH Calculate drag coefficient of a sphere using the Barati high-Re correlation (valid to Re=1E6).
CD_CEYLAN Calculate drag coefficient of a sphere using the Ceylan correlation.
CD_CHENG Calculate drag coefficient of a sphere using the Cheng correlation.
CD_CLIFT Calculate drag coefficient of a sphere using the Clift correlation.
CD_CLIFT_GAUVIN Calculate drag coefficient of a sphere using the Clift-Gauvin correlation.
CD_ENGELUND Calculate drag coefficient of a sphere using the Engelund-Hansen correlation.
CD_FLEMMER_BANKS Calculate drag coefficient of a sphere using the Flemmer-Banks correlation.
CD_GRAF Calculate drag coefficient of a sphere using the Graf correlation.
CD_HAIDER_LEV Calculate drag coefficient of a sphere using the Haider-Levenspiel correlation.
CD_KHAN_RICH Calculate drag coefficient of a sphere using the Khan-Richardson correlation.
CD_MIKHAILOV Calculate drag coefficient of a sphere using the Mikhailov-Freire correlation.
CD_MORRISON Calculate drag coefficient of a sphere using the Morrison correlation.
CD_MORSI_ALEX Calculate drag coefficient of a sphere using the Morsi-Alexander correlation.
CD_ROUSE Calculate drag coefficient of a sphere using the Rouse correlation.
CD_SONG_XU Calculate drag coefficient of a particle using the Song-Xu correlation for spherical and non-spherical particles.
CD_STOKES Calculate drag coefficient of a sphere using Stokes law (Cd = 24/Re).
CD_SWAMEE_OJHA Calculate drag coefficient of a sphere using the Swamee-Ojha correlation.
CD_TERFOUS Calculate drag coefficient of a sphere using the Terfous correlation.
CD_YEN Calculate drag coefficient of a sphere using the Yen correlation.
DRAG_SPHERE Calculate the drag coefficient of a sphere using various correlations based on Reynolds number.
SPHERE_FALL_DIST Calculate distance traveled by a falling sphere after a given time.
SPHERE_VEL_AT_T Calculate the velocity of a falling sphere after a given time.
TIME_V_TERMINAL Calculate time for a particle in Stokes regime to reach terminal velocity.
V_TERMINAL Calculate terminal velocity of a falling sphere using drag coefficient correlations.