Chapter 7, Problem 7.15P

(a)

Interpretation Introduction

Interpretation:

The minimum fluidization velocity for the particle is to be determined.

Concept Introduction:

The velocity at which the drag force of the upward moving gas is equal to the apparent weight of the particle is known as the minimum fluidization velocity. In other words, it is the superficial velocity at which the bed of particle is just fluidized.

The minimum fluidization velocity is obtained at the incipient fluidization.

At incipient fluidization,

  $\frac{\text{ΔP}}{\text{L}}\text{=}\text{g(1}\text{-}{\text{ε}}_{\text{M}}{\text{)(ρ}}_{\text{p}}\text{-}\text{ρ)}$ ...... (1)

The notations used here are,

  $\Delta \text{P}$ = Pressure drop

L = Length of bed

g = Acceleration due to gravity

  $\text{ρ}$ = Density of the fluid

  ${\text{ρ}}_{\text{p}}$ = Density of the particle

  ${\text{ε}}_{\text{M}}$ = Minimum bed porosity

The Ergun equation for the bed is given as,

  $\frac{\text{ΔP}}{\text{L}}\text{=}\frac{{\text{150<mover accent="true"><mo>V </mo><mo>¯</mo></mover>}}_{\text{0}}{\text{μ(1-ε)}}^{\text{2}}}{{\varphi }_{\text{s}}{}^{\text{2}}{\text{D}}_{\text{p}}{}^{\text{2}}{\text{ε}}^{\text{3}}}+\frac{\text{1}{\text{.75<mover accent="true"><mo>V </mo><mo>¯</mo></mover>}}_{\text{0}}{}^{\text{2}}\text{ρ(1-ε)}}{{\varphi }_{\text{s}}{\text{D}}_{\text{p}}{\text{ε}}^{\text{3}}}$ ...... (2)

The notations used here are,

  ${\text{<mover accent="true"><mo>V </mo><mo>¯</mo></mover>}}_0$ = Superficial velocity

D_p = Diameter of the particle

  ${\varphi }_s$ = Sphericity of the particle

  $\text{ε}$ = Bed porosity

  $\text{μ}$ = Fluid viscosity

Combine equation (1) and (2),

  $\text{g(1}\text{-}{\text{ε}}_{\text{M}}{\text{)(ρ}}_{\text{p}}\text{-}\text{ρ)}\text{=}\frac{{\text{150<mover accent="true"><mo>V </mo><mo>¯</mo></mover>}}_{\text{0M}}{\text{μ(1-ε}}_{\text{M}}{\text{)}}^{\text{2}}}{{\varphi }_{\text{s}}{}^{\text{2}}{\text{D}}_{\text{p}}{}^{\text{2}}{\text{ε}}_{\text{M}}{}^{\text{3}}}\text{+}\frac{\text{1}{\text{.75<mover accent="true"><mo>V </mo><mo>¯</mo></mover>}}_{\text{0M}}{}^{\text{2}}{\text{ρ(1-ε}}_{\text{M}}\text{)}}{{\varphi }_{\text{s}}{\text{D}}_{\text{p}}{\text{ε}}_{\text{M}}{}^{\text{3}}}$ ...... (3)

  ${\text{<mover accent="true"><mo>V </mo><mo>¯</mo></mover>}}_{\text{0M}}$ = Minimum fluidization velocity

The density of gas can be given as,

  $\text{ρ}\text{=}\frac{\text{PM}}{\text{RT}}\mathrm{.......}(4)$

In the equation (3), notations used are,

P is the pressure at which gas enters

M is the molecular weight of gas

R is the Universal Gas constant

T is the temperature at which gas enters

(b)

Interpretation Introduction

Interpretation:

For the given expansion of the bed the reason for the difficulty in prediction of the gas velocity is to be mentioned.

Concept Introduction:

The bed expansion is due to the space occupied by the gas bubbles.

In bubbling fluidization, the gas passes through the bed as bubbles or voids which is free from the solids.

The bed expansion can be given as,

  $\frac{\text{L}}{{\text{L}}_{\text{M}}}\text{=}\frac{{\text{u}}_{\text{b}}\text{-}{\text{<mover accent="true"><mo>V </mo><mo>¯</mo></mover>}}_{\text{0M}}}{{\text{u}}_{\text{b}}\text{-}{\text{<mover accent="true"><mo>V </mo><mo>¯</mo></mover>}}_{\text{0}}}$ ...... (1)

The notations used here are,

L = Bed height

L_M = Minimum bed height

  ${\text{u}}_{\text{b}}$ = Average bed velocity

  ${\text{<mover accent="true"><mo>V </mo><mo>¯</mo></mover>}}_0$ = Superficial velocity

  ${\text{<mover accent="true"><mo>V </mo><mo>¯</mo></mover>}}_{\text{0M}}$ = Minimum fluidization velocity

The bubble velocity is given as,

  ${\text{u}}_{\text{b}}\text{=}\text{0}\text{.7}\sqrt{{\text{gD}}_{\text{b}}}$ ...... (2)

g = Acceleration due to gravity

  ${\text{D}}_{\text{b}}$ = Bubble diameter