Filtration

Question 1

Darcy’s Law

Answer 1

u_c = K(-dP)/l

Question 2

Define Permeability constant

Answer 2

K = B/mu

Where B = permeability constant
So:
u_c = B(-dP)/(mu*l)

Purely a function of the cake

Question 3

Specific surface area

Answer 3

S = A/V

e.g for a sphere S = 6/d

Question 4

Specific bed area

Answer 4

Surface area presented to the fluid per volume of the bed

S_B = A/V_bed
= A/(V_p/(1-eps))
= A(1-eps)/V_p

V_p = Volume occupied by particles

Question 5

Carman-Kozeny Equation Theory

Answer 5

Assumes free space in powder bed consists of a series of tortuous channels
Assumes cake is uniform
Assumes voidage is constant (unlikely)

Question 6

Carman-Kozeny Equation Derivation

Answer 6

Start with:
u1 = dm’^2 (-dP)/(K’mu*l’)

• dm’ = eps/S_b
  = eps/(S*(1-eps))
• u1 = u_c/eps
• K’l’ = K’‘l

Insert those terms into equation and arrange for u_c

Insert K’’ = 5 (unless different)
Insert S = 6/d for a sphere

Question 7

What do the u_c, u1, dm’, and l’ terms mean in the Carman- Kozeny derivation

Answer 7

u_c = mean velocity over entire tube (Q/A)

u1 = mean velocity through tubes (Q/(eps*A))

dm’ = equivalent diameter of pore

l’ = channel length

Question 8

Give the equation for the permeability B from the Carman-Kozeny equation and Darcy’s Theory

Answer 8

B = eps^3 /(K’’ *S^2 * (1-eps)^2)

Question 9

Carman plot equation

Answer 9

R/(rou^2) = 5Re^-1 + 0.4*Re^-0.1

Question 10

Equation for the effect Tortuosity has on K’’

Answer 10

K’’ = (l’/l)^2 * K_0

Question 11

Equation for the correction factor due to the wall effect on K’’

Answer 11

fw = (1 + Sc/(2*S))^2

Question 12

Factors affecting K’’

Answer 12

• Tortuosity
• Wall effects (particles not packed as closely at the walls so resistance is not as great. Deviation is greater for smaller pipe diameters)
• Mixed sizes of particles such as filter aids can reduce packing & increase channel diameter
• High voidage(usually for non-spherical particels e.g. fibres/rings)

Question 13

Factors affecting the rate of filtration

Answer 13

• Pressure drop
• Filtering surface area
• Viscosity of filtrate
• Resistance of filter cake
• Resistance of filter medium

Question 14

What stage of cake formation is most important?

Answer 14

Initial phase:

• Rate of flow is greatest at the beginning (minimum resistance)
• High Initial rates may result in plugged pres (results in higher resistance)
• Orientation of particles in initial layer

Question 15

Incompressible cake equation

Answer 15

uc = 1/A dV/dt = (-dP)/(rmu*l)

r = 1/B
= specific resistance

r is taken to be constant

Question 16

Compressible cakes

Answer 16

1/A dV/dt = 1/(rmu) *(-dP/dz)

r = f(eps)
eps = f(z)

dV/dt = A^2/(muv’r) *(-dP/dV)

Question 17

Pretreatment of slurries before filtration

Answer 17

• Dilute solution

- Add filter aids