Terminal Settling Velocity of a Solid Particle in Fluid - Maple Help

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Terminal Settling Velocity of a Solid Particle in Fluid

 Introduction This application calculates the terminal velocity of a solid particle settling in a fluid.

Deriving the Settling Velocity

 > $\mathrm{restart}:$

Drag force

 > $\mathrm{Fd}≔\frac{1}{2}\cdot \mathrm{π}\cdot \frac{{\mathrm{Dia}}^{2}}{4}\cdot \mathrm{CD}\cdot \mathrm{ρ__f}\cdot {v}^{2}:$

Buoyancy force

 > $\mathrm{Fb}≔\left(\mathrm{ρ__p}-\mathrm{ρ__f}\right)\cdot g\cdot \mathrm{π}\cdot \frac{{\mathrm{Dia}}^{3}}{6}:$

The terminal settling velocity is reached when the drag force equals the buoyancy force. Hence, the settling velocity is given by the following equation.

 > $\mathrm{res}≔\mathrm{v__terminal}=\mathrm{solve}\left(\mathrm{Fb}=\mathrm{Fd},v\right)\left[1\right]$
 ${\mathrm{res}}{≔}\mathrm{v__terminal}{=}\frac{{2}{}\sqrt{{-}{3}{}{\mathrm{CD}}{}\mathrm{ρ__f}{}{g}{}{\mathrm{Dia}}{}\left(\mathrm{ρ__f}{-}\mathrm{ρ__p}\right)}}{{3}{}{\mathrm{CD}}{}\mathrm{ρ__f}}$ (2.1)

Parameters

Gravity:

 > $g≔9.81:$

Density of the particle:

 > $\mathrm{ρ__p}≔1800:$

Fluid density:

 > $\mathrm{ρ__f}≔994.6:$

Fluid viscosity:

 > $\mathrm{μ}≔0.0008931:$

Particle diameter:

 > $\mathrm{Dia}≔0.000208:$

Governing Equations

Drag coefficient:

 > $\mathrm{CD}≔\mathrm{Rey}\to {\begin{array}{cc}\frac{24}{\mathrm{Rey}}& \mathrm{Rey}<0.1\\ \frac{24}{\mathrm{Rey}}\left(1+0.14\cdot {\mathrm{Rey}}^{0.7}\right)& 0.1\le \mathrm{Rey}\le 1000\\ 0.44& 1000\le \mathrm{Rey}\le 350000\\ 0.19-\frac{8\cdot {10}^{4}}{\mathrm{Rey}}& \mathrm{Rey}>350000\end{array}:$
 >

 > $\mathrm{TerminalVelocity}≔\mathrm{v__terminal}=\sqrt{\frac{4g\left(\mathrm{ρ__p}-\mathrm{ρ__f}\right)\mathrm{Dia}}{3\mathrm{CD}\left(\mathrm{Rey}\right)\cdot \mathrm{ρ__f}}}:$

Solution

 > $\mathrm{fsolve}\left(\left\{\mathrm{ReynoldsNumber},\mathrm{TerminalVelocity}\right\},\left\{\mathrm{v__terminal},\mathrm{Rey}\right\}\right);$
 $\left\{{\mathrm{Rey}}{=}{3.656696458}{,}\mathrm{v__terminal}{=}{0.01578618582}\right\}$ (5.1)