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TRANSCRIPT
FILTRATION
���� Removal of solid particles from a fluid by passing the
fluid through a filtering medium, or septum.
MECHANSIMS OF FILTRATION
(a) Clarifiers
* Also known as “deep-bed filters”.
* The particles of solid are trapped inside the filter
medium.
(b) Cake filters
* The filter medium is relatively thin, compared with that
of a clarifying filter.
* After the initial period, the cake of solids does the
filtration, not the septum.
* A visible cake of appreciable thickness builds up on the
surface and must be periodically removed.
MECHANSIMS OF FILTRATION (3/3)
EQUIPMENT FOR CONVENTIONAL FILTRATION
(1) Plate and Frame Filter Press
* The most common type, but less common for
bioseparations.
* Used where a relatively dry cake discharge is desired.
* Cake removal: open the whole assembly
���� Should not be used where there are toxic fumes or
biohazards.
(2) Horizontal Plate Filter:
* Filtration occurs from the top of each plate.
* Cake removal: removed with a sluicing nozzle or
discharged by rapidly rotating the leaves.
EQUIPMENT FOR CONVENTIONAL FILTRATION (2/7)
(3) Vertical Leaf Filter and Candle Type Vertical Tank
Filter:
EQUIPMENT FOR CONVENTIONAL FILTRATION (3/7)
(3) Vertical Leaf Filter and Candle Type Vertical Tank Filter (2/2):
* Have a relatively high filtration area per volume.
���� Require only a small floor area.
* Filter cake is formed on the external surface of the tubes.
* The tubes are cleaned by backwashing.
EQUIPMENT FOR CONVENTIONAL FILTRATION (4/7)
(4) Rotary Vacuum Filter:
* Rotate at a low speed during the operation.
* Pressure inside the drum is a partial vacuum.
���� Liquid is sucked through the filter cloth and solids
are retained on the surface of the drum.
* Three chief steps of the filtration cycle:
(1) cake formation
(2) cake washing (to remove either valuable or unwanted
solutes)
(3) cake discharge
EQUIPMENT FOR CONVENTIONAL FILTRATION (6/7)
* The workhorse of bioseparations.
* Common for large-scale operations whenever the solids
are difficult to filter.
* Being automated.
���� Have a lower labor cost.
EQUIPMENT FOR CONVENTIONAL FILTRATION (7/7)
PRETREATMENT OF FILTRATION
““““Filtration is a straightforward procedure for well-defined crystals.”
* Fermentation beers and other biological solutions are
notoriously hard to filter, because of: (1) high, non-
newtonian viscosity, and (2) highly compressible filter cakes.
���� Conventional filtration is often too slow to be practical.
���� The filtration requires pretreatment: heating,
coagulation and flocculation, or adsorption on
filter aid.
A. Heating
* To improve the feed’s handling characteristics.
(Thinking of filtering a dilution solution of egg white.)
* The simplest pretreatment (and the least expensive).
* Chief constraint: thermal stability of the product.
PRETREATMENT OF FILTRATION (2/12)
B. Coagulation and Flocculation
* Through the addition of electrolytes.
* Types of coagulants:
(1) Simple electrolytes (such as ferric chloride, alum,
or acids and bases)
(2) Synthetic polyelectrolytes
PRETREATMENT OF FILTRATION (3/12)
coagulation flocculation
* Action of simple electrolytes: reduce the electrostatic
repulsion existing between colloidal particles.
* Action of synthetic polyelectrolytes:
(1) Reduce electrostatic repulsion
(2) Adsorb on adjacent particles
* Commercially available polyelectrolytes (can be anionic,
cationic, or nonionic): polyacrylamides, polyethylenimines,
and polyamine derivatives.
PRETREATMENT OF FILTRATION (4/12)
C. Adsorption on Filter Aids
* Why filter-aid filtration?
���� Two major problems can be reduced:
(1) High compressibility of the accumulated
biomass
(2) Penetration of small particles into the filter
medium
���� Lengthen the filtration cycle; improve
the quality of the filtered liquor.
PRETREATMENT OF FILTRATION (6/12)
* The effect of
filter aid on
filtrate volume
for Streptomyces
griseus:
PRETREATMENT OF FILTRATION (7/12)
* The effect of pH and filter aid on filtrate volume for
Streptomyces griseus:
PRETREATMENT OF FILTRATION (8/12)
* How does the filter-aid help?
(1) Give porosity to the filter cake.
Solids to be filtered Porosity
Hard spheres of the same size ≈≈≈≈ 0.45
General cases 0.2−−−−0.3
Compressible solids ≈≈≈≈ 0
Diatomaceous silica (矽藻土矽藻土矽藻土矽藻土) ≈≈≈≈ 0.9
(2) Create a very large surface to trap the gelatinous
precipitate.
���� Allow much more filtrate to be obtained before
eventually clogging up.
PRETREATMENT OF FILTRATION (9/12)
���� - Protect the filter medium from fouling.
- Provide a finer matrix to exclude particles
from the filtrate.
* How to use filter-aid?
(1) Precoat—a thin layer (0.1 to 0.2 lb/ft2) of filter aid is
deposited on the filter medium prior to introducing
the filter feed to the system
(2) Body feed—add the filter aid to the filter feed
PRETREATMENT OF FILTRATION (10/12)
* The use of filter-aid is mainly for removing small
amounts of unwanted particulate material.
���� It cannot deal with large quantities of precipitate
successfully.
* Types of filter-aid (the most effective):
(1) Diatomaceous earths such as Celite (consisting mainly of SiO2)
(2) Perlites (volcanic rock processed to yield an expanded form)
���� Note: some products like the aminoglycoside antibiotics
may irreversibly bind to diatomaceous earth.
PRETREATMENT OF FILTRATION (12/12)
GENERAL THEORY FOR FILTRATION
Darcy’s law—relate the flow rate through a porous bed of
solids to the pressure drop causing that flow.
lµPk
v∆
=
v = velocity of the liquid
∆∆∆∆P = pressure drop across the bed of thickness ℓ
∆∆∆∆P/ ℓ = pressure gradient
µµµµ = viscosity of the liquid
k = permeability of the bed, a proportionality
constant (dimension: L2)
* Like Ohm’s law, ℓ/k is the resistance of filtration.
Strictly speaking, Darcy’s law holds only when
5)1(
<− εµ
ρvd
where d is the particle size of the filter cake, ρρρρ is the liquid density, and εεεε is the void fraction in the cake.
* Biological separations almost always obey this inequality.
For a batch filtration,
dt
dV
Av
1=
lµPk
dt
dV
A
∆=
1�
where V is the total volume of filtrate, A is the filter
area, and t is the time.
GENERAL THEORY FOR FILTRATION (2/5)
lµPk
v∆
=Darcy’s law:
Two contributions to the filtration resistance:
CM RRk
+=l
where RM is the resistance of the filter medium (constant),
and RC is the resistance of the cake (varies with V).
The basic differential equation for filtration at constant
pressure drop can thus be obtained as:
)(
1
CM RR
P
dt
dV
A +∆
=µlµ
Pk
dt
dV
A
∆=
1����
GENERAL THEORY FOR FILTRATION (3/5)
Incompressible Cakes
αααα = specific cake resistance, cm/g
ρρρρ0 = mass of cake solids per volume of filtrate
)(
1
CM RR
P
dt
dV
A +
∆=
µ
])/([
1
0 MRAV
P
dt
dV
A +∆
=αρµ� (I.C.: t = 0, V = 0)
�
=A
VRC 0αρ
BA
VK
P
R
A
V
PV
At M +
=
∆+
∆
=µµαρ
2
0
GENERAL THEORY FOR FILTRATION (4/5)
Plot
V
Atversus
A
V� Slope =
PK
∆=
2
0µαρ
Known µµµµ, ρρρρ0, ∆∆∆∆P ���� αααα can be determined.
* Often, the medium resistance RM is insignificant, B = 0.
2
0
2
∆
=A
V
Pt
µαρ
BA
VK
P
R
A
V
PV
At M +
=
∆+
∆
=µµαρ
2
0
GENERAL THEORY FOR FILTRATION (5/5)
[Example] A suspension containing 225 g of carbonyl iron
powder, Grade E, per liter of a solution of 0.01 N NaOH is
to be filtered, using a leaf filter. Estimate the size (area) of
the filter needed to obtain 100 lb of dry cake in 1 h of
filtration at a constant pressure drop of 20 psi. The cake is
incompressible. The specific cake resistance is 1011 ft/lb.
The resistance of the medium is taken as 0.1 in−−−−1.
P
R
A
V
PV
At M
∆+
∆=
µµαρ2
0
Solution:
3
30lb/ft 0.14
g 453.6
lb
ft
L32.28
L
g 225
filtrate of volume
solid cake of mass=
==ρ
3
3ft 1.7
lb/ft 14.0
lb 100filtrate of volume ===V
(To be continued)
[Example] A suspension containing 225 g of carbonyl iron powder, Grade E, per
liter of a solution of 0.01 N NaOH is to be filtered, using a leaf filter. Estimate the
size (area) of the filter needed to obtain 100 lb of dry cake in 1 h of filtration at a
constant pressure drop of 20 psi. The cake is incompressible. The specific cake
resistance is 1011 ft/lb. The resistance of the medium is taken as 0.1 in-1.
P
R
A
V
PV
At M
∆+
∆
=µµαρ
2
0Solution (cont’d):
t = filtration time = 1 h
×=∆
2
22
2
f
2
f
3
h
s )3600(
s-lb
ft-lb2.32
psi 14.7
/ftlb 10116.2 psi 20P
= 1.2 ×××× 1012 lb/ft-h2
αααα = specific cake resistance = 1011 ft/lb
RM = resistance of the medium = 0.1 in−−−−1 = 1.2 ft−−−−1
µµµµ = viscosity of the liquid = 1 cp = 2.42 lb/ft-h (assumed)
(To be continued)
Solution (cont’d):
P
R
A
V
PV
At M
∆+
∆
=µµαρ
2
0
1212
11
102.1
)2.1)(42.2(1.7
)102.1(2
)0.14)(10)(42.2(
1.7
)1(
×+
×=
A
A����
A2 −−−− 1.7 ×××× 10-11A −−−− 71.2 = 0
A = 8.4 ft2
#
[Example] A suspension containing 225 g of carbonyl iron powder, Grade E, per
liter of a solution of 0.01 N NaOH is to be filtered, using a leaf filter. Estimate the
size (area) of the filter needed to obtain 100 lb of dry cake in 1 h of filtration at a
constant pressure drop of 20 psi. The cake is incompressible. The specific cake
resistance is 1011 ft/lb. The resistance of the medium is taken as 0.1 in-1.
[Example] Streptomyces Filtration from an Erythromycin Broth.
Using a test filter, we find the following data for a broth
containing the antibiotic erythromycin and added filter aid:
The filter leaf has a total area of 0.1 ft2 and the filtrate has a
viscosity of 1.1 cp. The pressure drop is 20 in. of mercury and
the feed contains 0.015 kg dry cake per liter. Determine the
specific cake resistance αααα and the medium resistance RM.
P
R
A
V
PV
At M
∆+
∆
=µµαρ
2
0Solution:
(To be continued)
Example: Streptomyces Filtration from an Erythromycin Broth (cont’d)
#
P
R
A
V
PV
At M
∆+
∆
=µµαρ
2
0
[Example] We have filtered a slurry of sitosterol at constant
pressure through a filtration medium consisting of a screen
support mounted across the end of a Pyrex pipe. We find
that the resistance of the filtration medium is negligible. We
also find the following data in a laboratory test:
On the basis of this laboratory test, predict the number of
frames (30 in ×××× 30 in ×××× 1 in thick) needed for a plate-and-frame press. Estimate the time required for filtering a 63 kg
batch of steroid. In these calculations, assume that the feed
pump will deliver 10 psi and that the filtrate from the press
must be raised against the equivalent of 15 ft head.
(To be continued)
Example: filtering a slurry of sitosterol
Solution (cont’d):
(a) Predict the number of frames needed
3
3g/cm 245.0
cm 3.253
g 62 density Cake ==
Cake volume of 63 kg steroid = 35
3
3
cm 1057.2g/cm 0.245
g 1063×=
×
Number of frames needed = 4.17cm 2.54
in
in 13030
cm 1057.23
3
35
=
××
×
���� 18 frames are needed.
(To be continued)
(b) Time required for filtering a 63 kg batch of steroid
Solution (cont’d):
For incompressible cake with a negligible filter
medium resistance,
2
0
0
2
0 1
2or
2
∆
=
∆
=A
V
Pt
A
V
Pt
ρρ
µαµαρ
In the laboratory test:
2
20 cm) 08.5(4
g 62
psi) 15( 2 min 163
=πρ
µα2
4
0 g
cm-psi-min 261
2=
ρµα
����
Example: filtering a slurry of sitosterol
(To be continued)
(b) Time required for filtering a 63 kg batch of steroid (cont’d)
Solution:
In the large-scale operation:
25
2
2 cm 1009.2in
cm 54.2 in )3030(218 ×=
×××=A
psi 5.3(water) headft 33.9
psi 7.14 headft 15 psi 10 =
−=∆P
min 8.61009.2
000,63
5.3
1261
1
2
2
5
2
0
0
=
×××=
∆
=A
V
Pt
ρρ
µα
Example: filtering a slurry of sitosterol
#
In the laboratory test: 2
4
0 g
cm-psi-min 261
2=
ρµα
Compressible Cakes
““““Almost all cakes formed of biological materials are compressible. As these cakes compress, filtration
rates drop.”
To estimate the effects of compressibility, we assume that
the cake resistance αααα is a function of the pressure drop.
sP)(' ∆= αα
where αααα’ = a constant related largely to the size and shape
of the particles forming the cake
s = the cake compressibility
GENERAL THEORY FOR FILTRATION: Compressible Cakes (1/3)
=A
VRC 0αρRecall:
sP)(' ∆= αα Ps ∆+= log'loglog αα
���� Plot logαααα versus log∆∆∆∆P, slope = s, intercept = logαααα’.
����
GENERAL THEORY FOR FILTRATION: Compressible Cakes (2/3)
sP)(' ∆= αα
•••• For a rigid, incompressible cake, s = 0.
•••• For a highly compressible cake, s ≈≈≈≈ 1.
•••• In practice, s ranges from 0.1−−−−0.8.
•••• When values of s are high, one should consider pretreating the feed with filter aids.
GENERAL THEORY FOR FILTRATION: Compressible Cakes (3/3)
=A
VRC 0αρRecall:
[Example] Filtration of Beer Containing Protease. We have a
suspension of Bacillus subtilis fermented to produce the enzyme
protease. To separate the biomass, we have added 1.3 times the
biomass of a Celatom filter aid, yielding a beer containing 3.6
wt% solid, with a viscosity of 6.6 cp. With a Buchner funnel 5
cm in diameter attached to an aspirator, we have found that we
can filter 100 cm3 of this beer in 24 min. However, previous
studies with this type of beer have had a compressible cake with
s equal to 2/3.
We now need to filter 3000 L of this material in a pilot
plant’s plate-and-frame press. This press has 15 frames, each
of area 3520 cm2. The spacing between these frames can be
made large, so that we can filter all the beer in one single run.
The resistance of the filter medium is much smaller than the
filter cake, and the total pressure drop that can be used is 65 psi.
How long will it take to filter this beer at 50 psi?
(To be continued)
Example: Filtration of Beer Containing Protease
Solution (cont’d):
Negligible RM ����
2
0
2
∆
=A
V
Pt
µαρ
Compressible cake, s
P)(' ∆= αα2
1
0
2
'
∆
=− A
V
Pt
s
ρµα����
Laboratory test:
∆∆∆∆P = 14.7 psi (a Buchner funnel attached to an aspirator)
A = ;V = 100 cm3; t = 24 min; s = 2/3 2)cm 5(4
π
2
2
3
3/1
0
)cm 5(4
cm 100
)psi 7.14(2
' min 24
=π
ρµα
���� µαµαµαµα’ρρρρ0 = 4.53 min psi1/3 cm−−−−2
����
(To be continued)
2
1
0
2
'
∆
=− A
V
Pt
s
ρµαµαµαµαµα’ρρρρ0 = 4.53 min psi
1/3 cm−−−−2
Pilot-plant operation:
V = 3000 L = 3 ×××× 106 cm3
A = 15 ×××× 2 ×××× 3520 cm2 (Filtration occurs on both sides of the frame.)
26
3/1
2
1
0
3520215
103
)50(2
53.4
2
'
××
×=
∆
=− A
V
Pt
s
ρµα
;
= 496 min = 8.3 h
Example: Filtration of Beer Containing Protease
Solution (cont’d):
#
ANALYSIS OF CONTINUOUS ROTARY
VACUUM FILTERS
There are three
stages involved in the
operation:
(1) cake formation
(2) cake washing
(3) cake discharge
(not affecting
the filter size
and the cycle
time)
Cake Formation
For compressible cake and negligible medium resistance,
2
1
0
2
1
0
2
'or
2
'
∆
=
∆= −−
A
V
Pt
A
V
Pt
f
sfs
ρµαρµα
where tf = cake formation time
Vf = volume of filtrate collected during the period
of tf
A = filtration area (submerged area of filter)
ANALYSIS OF CONTINUOUS ROTARY VACUUM FILTERS (2/8)
Cake Formation (cont’d)
2
1
0
2
'
∆
= −A
V
Pt
f
sf
ρµα
Let tf = ββββtc and A = ββββAT
2
1
0
2
'
∆=
−T
f
scA
V
Pt
βρµα
β
where tc = cycle time
AT = total filter area
ββββ = fraction of the drum submerged
����
ANALYSIS OF CONTINUOUS ROTARY VACUUM FILTERS (3/8)
Cake Washing
Two factors involved in the stage of cake washing:
(1) The fraction of soluble material remained after the wash
���� Governing the volume of wash liquid required.
(2) The rate of wash liquid passes through the cake
���� Controlling the fraction of cycle time for cake
washing.
ANALYSIS OF CONTINUOUS ROTARY VACUUM FILTERS (4/8)
An empirical equation for the fraction of soluble material
remained: nr )1( ε−=
where r = ratio of soluble material remained after the
wash to that originally present in the cake
n = volume of wash liquid divided by the volume
of retained liquid
εεεε = washing efficiency of the cake
Two factors involved in the stage of cake washing:
(1) The fraction of soluble material remained after the wash
���� Governing the volume of wash liquid required.
ANALYSIS OF CONTINUOUS ROTARY VACUUM FILTERS (5/8)
The wash liquid contains no additional
solids.
���� (1) The cake thickness is constant.
Two factors involved in the stage of cake washing:
(2) The rate of wash liquid passes through the cake
���� Controlling the fraction of cycle time for cake washing.
(2) Wash rate
= filtration rate at the end of cake formation
���� The flow of wash liquid is
constant.
ANALYSIS OF CONTINUOUS ROTARY VACUUM FILTERS (6/8)
w
ww
At
V
dt
dV
A==
1 rate Wash
where Vw = volume of wash water required, and tw = time
required for washing.
Filtration rate at the end of cake formation = ftt
dt
dV
A =
1
2/1
0
12
1
0
'
)(2or
2
'
∆=
∆
=−
− ρµαρµα tP
A
V
A
V
Pt
s
s
2/1
0
1
'2
)(1 rate Wash
∆=
==
−
== f
s
ttttt
P
A
V
dt
d
dt
dV
Aff
ρµα
2/1
0
1
'2
)(
∆=∴
−
f
s
w
w
t
P
At
V
ρµα
ANALYSIS OF CONTINUOUS ROTARY VACUUM FILTERS (7/8)
A useful expression:
2/1
0
12/1
0
1
'
)(2 and
'2
)(
∆=
∆=
−−
f
s
f
f
f
s
w
w
t
P
At
V
t
P
At
V
ρµαρµα
nfV
V
V
V
V
V
t
t
f
r
r
w
f
w
f
w 222 ===����
2/1
1
0
2/1
1
0
)(2
' and
)(
'2
∆=
∆=
−− s
ff
fs
fw
wP
t
A
Vt
P
t
A
Vt
ρµαρµα����
where Vr = volume of liquid retained
f = ratio of the volume of retained liquid (Vr) to
the volume of filtrate (Vf)
ANALYSIS OF CONTINUOUS ROTARY VACUUM FILTERS (8/8)
[Example] It is desired to filter a cell broth at a rate of
2000 L/h on a rotary vacuum filter at a vacuum pressure of
70 kPa. The cycle time for the drum is 60 s, and the cake
formation time is 15 s. The broth to be filtered has a
viscosity of 2.0 cp and a cake solid per volume of filtrate of
10 g/L. From laboratory tests, the specific cake resistance
has been determined to be 9 ×××× 1010 cm/g. Determine the area of the filter that is required.
Solution:
For incompressible cake,
Pt
VA
A
V
Pt
f
ff
f ∆=
∆=
2or
2
2
02
2
0µαρµαρ
(To be continued)
Example: Determine the area of a rotary vacuum filter
Solution (cont’d):
s-cm
g 0.02 cp 2 ==µ
g
cm 109
10×=α 3
3
0cm
g 1010
L
g 10
−×==ρ; ;
33
3 cm 8333s 3600
h s) 15(
h
cm 102000 =
×
×=
fV
2
5
22
3
s-cm
g 100.7
cm 100
m
kg
g 1000
s-N
m-kg
m
N 1070 kPa 70 ×=
×==∆P
47
5
23102
02 cm 1095.5)100.7)(15(2
)8333)(1010)(109)(02.0(
2×=
×××
=∆
=−
Pt
VA
f
fµαρ
���� A = 7715 cm2 = 0.7715 m2
2
Tm 09.3
15
607715.0 =×=×=
f
c
t
tAA����
#
[Example] We want to filter 15,000 L/h of a beer containing
erythromycin using a rotary vacuum filter originally
purchased for another product. Our filter has a cycle time
of 50 s and an area of 37.2 m2. It operates under a vacuum
of 20 in Hg. The pretreated broth forms an incompressible
cake with the resistance:
20 s/cm 292
=∆P
µαρ
We want to wash the cake until only 1% of the retained
solubles is left, and we expect that the washing efficiency
will be 70% and that 1% of the filtrate is retained. (a)
Calculate the filtration time per cycle. (b) Find the washing
time.
Solution:
For incompressible cake,
2
0
2
∆==
T
f
cfA
V
Ptt
βµαρ
β
(To be continued)
Example: Filtration of erythromycin using rotary vacuum filter
Solution (cont’d):
For incompressible cake, 2
0
2
∆==
T
f
cfA
V
Ptt
βµαρ
β
tc = 50 s
AT = 37.2 m2 = 37.2 ×××× 104 cm2
20 s/cm 292
=∆P
µαρ
����
33 cm 10208 L208s 3600
h)s 50() L/h000,15( ×==
××= βββ
fV
s 1.9102.37
1020829
2 4
32
0 =
×××
=
∆=
ββ
βµαρ
T
f
fA
V
Pt
(To be continued)
Solution (cont’d):
n
f
w rnft
t)1( and 2 ε−==
Fraction of retained solubles, r = 0.01
Washing efficiency, εεεε = 0.7
Fraction of filtrate retained, f = 0.01
r = 0.01 = (1 −−−− 0.7)n ���� n = 3.82
tw = 2nf ×××× tf = 2 ×××× 3.82 ×××× 0.01 ×××× 9.1 = 0.7 s
Example: Filtration of erythromycin using rotary vacuum filter
(b) Find the washing time.
#
Application of Rotary Vacuum Filter
* It is commonly used to recover yeast and mycelia.
* Filtration of bacterial fermentation broth will usually
require a precoat of filter aid.
* The separation of cell debris is performed by adding
filter aid to the feed liquor.
CENTRIFUGAL
FILTRATION
* A combination of a
centrifuge and a filter.
* Accumulated solids
can be washed.
Hydrostatic Equilibrium in a Centrifugal Field
In a rotating centrifuge, a layer of liquid is thrown outward
from the axis of rotation and is held against the wall of the
bowl by centrifugal force.
CENTRIFUGAL FILTRATION (3/8)
Consider a volume element of
thickness dr at a radius r,
dmrdF 2ω= drrhdm )2( πρ=;
dF = centrifugal force
dm = mass of liquid in the element
ωωωω = angular velocity
ρρρρ = density of the liquid h = height of the ring
rdrrh
dFdPdrrhdF 222
2 and 2 ρω
πωπρ ==−=∴
Integration ���� )(2
1 2
1
2
2
2
21rrPPP −=∆−=− ρω
CENTRIFUGAL FILTRATION (4/8)
Principles of Centrifugal Filtration
R1 = radius of the surface of feed
solution
Rc = radius of the cake’s interface
Darcy’s law:
vk
PPkv µ
µ1
or =∆∆
=ll
Set 0
1αρ=
kv
P0µαρ=
∆l
����
For centrifugal filtration, the
pressure drop varies with the
radius, thus
vdr
dP0µαρ=−
CENTRIFUGAL FILTRATION (5/8)
vdr
dP0
µαρ=−
The total volumetric flow rate, Q = (2ππππrh)v; or rh
Qv
π2=
(Note: v varies with r.)
=−∴rh
Q
dr
dP
πµαρ
2
0
Integration ���� cR
R
h
QP 0
0 ln2
=∆−
πµαρ
The pressure drop (−−−−∆∆∆∆P) is due to the centrifugal force on
the liquid.
)(2
1 2
1
2
0
2 RRP −=∆− ρω)/ln(
)(
0
2
1
2
0
0
2
cRR
RRhQ
−=
µαρρωπ
����
* Note: Rc is a function of time, and so is Q; however, Q is
not a function of r.
CENTRIFUGAL FILTRATION (6/8)
)/ln(
)(
0
2
1
2
0
0
2
cRR
RRhQ
−=
µαρρωπ
Mass balance for the solids:
hRRV )( 2
c
2
0c0 −= πρρ (where ρρρρc = cake density)
)/ln(
)()2(
0
2
1
2
0
0
2
0 c
c
c
c
RR
RRh
dt
dRR
h
dt
dVQ
−=−==
µαρπρω
ρπρ
)/ln(
1
2
)(
0
2
1
2
0
2
ccc
c
RRR
RR
dt
dR
µαρρω −
=−����
I. C.: t = 0, Rc = R0
CENTRIFUGAL FILTRATION (7/8)
)/ln(
1
2
)(
0
2
1
2
0
2
ccc
c
RRR
RR
dt
dR
µαρρω −
=−
I. C.: t = 0, Rc = R0
The integrated expression is complex,
and can be approximated as:
−−
−=
cc
cc
R
R
R
R
RR
Rt 0
2
0
21
20
2
2
ln21)(2ρω
µαρ
���� This is the desired result to find the time needed for
obtaining a cake of thickness (R0 −−−− Rc).
* Recalling that for a flat cake, 2
0
2
∆=
A
V
Pt
µαρ
CENTRIFUGAL FILTRATION (8/8)
[Example] We can filter 250 cm3 of a slurry, containing
0.016 g progesterone (黃體激素黃體激素黃體激素黃體激素) per cm3, in 32 min. Our filter
has a surface area of 8.3 cm2, a pressure drop of 1 atm, and
a filter medium of negligible resistance. The solids in the
cake have a density of 1.09 g/cm3, and the slurry density is
that of water.
We want to use this experiment to estimate the time to
filter 1,600 liters of this slurry through a centrifugal filter.
The filter has a basket of 51 cm radius and 45 cm height. It
rotates at 530 rpm. When it is spinning, the liquid and cake
together are 5.5 cm thick. How long will this filtration take?
Solution:
−−
−=
cc
cc
R
R
R
R
RR
Rt 0
2
0
2
1
2
0
2
2
ln21)(2ρω
µαρ
���� Need data of µαµαµαµα and Rc.
(To be continued)
[Example] We can filter 250 cm3 of a slurry, containing 0.016 g
progesterone (黃體激素黃體激素黃體激素黃體激素) per cm3, in 32 min. Our filter has a surface area of 8.3 cm2, a pressure drop of 1 atm, and a filter medium of
negligible resistance. The solids in the cake have a density of 1.09 g/cm3,
and the slurry density is that of water.
Example: filtration of progesterone (2/3)
Solution (cont’d):
In the laboratory test,
2
0
2
∆
=A
V
Pt
µαρ
t = 32 min = 1920 s; ρρρρ0 = 0.016 g/cm3
2
6226
s-cm
g 1001.1
dyne
cm/s-g
atm
dyne/cm 1001.1 atm 1 ×=
×=∆P
V = 250 cm3; A = 8.3 cm2
2
6 3.8
250
)10(1.012
) 016.0( 1920
×
=µα
���� ���� µαµαµαµα = 2.67 ×××× 108 s-1
(To be continued)
Using centrifugal filtration,
−−
−=
cc
cc
R
R
R
R
RR
Rt 0
2
0
21
20
2
2
ln21)(2ρω
µαρ
µαµαµαµα = 2.67 ×××× 108 s-1 ; ρρρρc = 1.09 g/cm3 ; ρρρρ = 1.0 g/cm3
ωωωω = 530 rpm = 55.47 s-1 ; R0 = 51 cm ;
R1 = 51 −−−− 5.5 = 45.5 cm
Mass balance for solids: hRRV cc )( 22
00 −= πρρ
���� (0.016)(1,600 ×××× 103) = (1.09)ππππ[(51)2 −−−− Rc2](45)
���� Rc = 49.3 cm
s 4663.49
51ln21
3.49
51
)5.4551()47.55)(0.1(2
)3.49)(09.1)(1067.2(
2
222
28
=
−−
−
×=∴ t
Solution (cont’d):
#
Example: filtration of progesterone (3/3)