Sedimentation

Question 1

Define R’

Answer 1

R’ is the force per unit projected area of the particles

R’ = F/(pi*d^2 /4)

Question 2

Drag force correlation graphs for R’/(ro*u^2)

Answer 2

• Region a (12*Re^-1)
• Region b (12*Re^-1 + 0.22)
• Region c (0.22)
• Region d (0.05)

Question 3

Method of calculating the drag force

Answer 3

• Take value of R’/(ro*u^2) from graph
• Insert R’ = 4F/pid^2)
• Rearrange for F

Example for Laminar region is Stokes Flow
F = 3pimuud

Question 4

Calculation of terminal velocity of particle

Answer 4

• Take gravitational weight:
  Fg = ro_sVg
• Take buoyancy:
  Fb = ro_fVg
• Take value calculated for Fd from R’/(ro*u^2) graph
• Balance forces (Fd will act in opposite direction to motion)
• Rearrange for u

(By using V you assume particle is fully submerged)

Question 5

Assumptions for the validity of terminal velocities

Answer 5

• free settling (not hindered by the presence of other particles)
• No retarding effects from walls
• Fluid is continuous medium (particle is much larger than the free path of molecules)

Question 6

7 other effects on the falling velocity

Answer 6

• Rising velocity (rising particles will generate vortices and slow falling ones)
• Effect on boundaries
• Fine particles
• Fluid trubulence
• Fluid Motion
• Non-spherical particles
• Concentrated solutions

Question 7

How do rising velocities affect falling velocities

Answer 7

Rising particles generate vortices that slow falling ones

Question 8

How do boundaries affect falling velocities

Answer 8

• Displacement of liquids as particles settle
• Density of suspension (affects the buoyancy force)
• Flow pattern of liquid

Question 9

How does the particles being a fine particle affect its falling velocity

Answer 9

• Very fine particles are affected by natural convection and brownian motion
• Can also slip and so achieve higher velocities

Question 10

How does the fluid turbulence affect the falling velocity

Answer 10

Eddies formed increase the coefficient of friction and so slow the falling particle

Question 11

5 factors affecting sedimentation

Answer 11

• Height of suspension
• Diameter of Vessel (smaller diameters will have retardation effects)
• Concentration of suspension (higher C = lower rate)
• Shape of Vessel (inclined walls will have an impact)
• Flocculation/Coagulation

Question 12

How to calculate the estimation of dp for a non-spherical particle

Answer 12

• Calculate the projected Area, A
• Set A = pi*dp^2 /4
• Solve for dp

Question 13

Calculating the volume shape factor

Answer 13

• Calculate estimation for dp

- k’ = V/dp^3

Question 14

Solving for terminal velocity for a non spherical particle

Answer 14

• Calculate estimation for dp
• Calculate estimation for k’
• Balance forces for Fd (N.B. V = k’*dp^3)
• Set Fd = R’ * pi*dp^2
• Divide both sides by ro*u^2 and multiply by Re^2
• Use tables to give Re’
• Use tables to find adjustment to Re’ based on value for k’
• Take final Re’ value and rearrange for the terminal velocity

Question 15

When deos flotation occur

Answer 15

when ro_s < ro_f

Question 16

Centrifugation derivation

Answer 16

• Balance “gravitational” forces with buoyancy forces as for sedimentation
• instead of g use (omega^2 *r)
• Assume Fd is that for stokes Law but instead of u0 we have dr/dt
• insert u0 term from previous sedimentation theory
  (u0 = d^2 (ro_s - ro)g/(18gmu)

Question 17

Is there a terminal velocity in centrifugation

Answer 17

• Not as such.

- It is equivalent to an instantaneous terminal velocity in a gravitational velocity raised by a factor r*omega^2 /g