Chapter 7, Problem 7.9P

(a)

Interpretation Introduction

Interpretation:

The terminal velocity of the urea pellets for the free settling is to be calculated.

Concept Introduction:

The terminal velocity is the constant velocity which a particle attains when moving in a fluid at the zero acceleration.

The settling regime criteria determine the type of flow and the range in which the particle lies. The equation for the same is,

  $\text{K}\text{=}{\text{D}}_{\text{p}}{\left[\frac{{\text{gρ(ρ}}_{\text{p}}\text{-}\text{ρ)}}{{\text{μ}}^{\text{2}}}\right]}^{\frac{\text{1}}{\text{3}}}$ ...... (1)

The notations used are,

K = Settling criteria constant

D_p = Diameter of the particle

g = Acceleration due to gravity

  ${\text{ρ}}_{\text{p}}$ = Particle Density

  $\text{ρ}$ = Fluid density

  $\text{μ}$ = Fluid viscosity

For Stokes’ law regime, K < 2.6

For very small Reynolds number, Stokes law is applied to calculate the terminal velocity and equation for the same is given as,

  ${\text{u}}_{\text{t}}\text{=}\frac{{\text{gD}}_{\text{p}}{}^{\text{2}}{\text{(ρ}}_{\text{p}}\text{-ρ)}}{\text{18μ}}\mathrm{.......}\text{(2)}$

For Newton’s law regime, 68.9 < K < 2360

For large Reynolds number, Newton’s law is applied to calculate the terminal velocity and equation for the same is given as,

  ${\text{u}}_{\text{t}}\text{=}\text{1}\text{.75}\sqrt{\frac{{\text{gD}}_{\text{p}}{\text{(ρ}}_{\text{p}}\text{-ρ)}}{\text{ρ}}}\mathrm{.......}\text{(3)}$

(b)

Interpretation Introduction

Interpretation:

Velocity of the pellets at the bottom of the tower is to be determined and compare it with the terminal velocity calculated in previous part.

Concept Introduction:

There are 3 forces which act on a particle when it is present in the fluid. These forces are, upward buoyant force, drag force and gravity force. The equation which is obtained on combining the 3 forces are,

  $\frac{\text{du}}{\text{dt}}\text{=}\text{g}\frac{{\text{(ρ}}_{\text{p}}\text{-}\text{ρ)}}{{\text{ρ}}_{\text{p}}}\text{-}\frac{{\text{C}}_{\text{D}}{\text{u}}^{\text{2}}{\text{ρA}}_{\text{p}}}{\text{2m}}$ ...... (1)

The notations used are,

m = Mass of the particle

g = Acceleration due to gravity

  ${\text{ρ}}_{\text{p}}$ = Particle Density

  $\text{ρ}$ = Fluid density

C_D = Drag coefficient

u = Velocity of particle relative to fluid

  $\frac{\text{du}}{\text{dt}}$ = Acceleration of the particle

A_p = Projected area of particle