THIN-CAKE FILTRATION THEORY

A. Introduction
In most industrial solid-liquid separation applications, the process steps of filtration, washing and
dewatering (drying) are conducted on one filter. This paper describes the thin-cake filtration
theory for these operations.
B. Filtration
During the initial mechanism of cake forming in the filtration step, the filter cloth acts just to
initiate filtration by capturing the first particles. These first particles form bridges over the pores
of the cloth (bridging). During this initial phase, smaller particles may pass through the filter
cloth leading to turbid liquid (filtrate). As soon as a first layer of particles has accumulated on the
filter medium, this cake will then act as the actual filter medium.
When plotting the amount of filtrate obtained at constant filtration pressure versus the filtration
time, you get a filtration curve, which represents a root function, VF = Constant * tF
0.5
, as shown
in Figure 1. (Data, in Figure 1, is from an actual BHS test installation).
Filtrate versus filtration time on a
BHS-AP300/1 (filter area 0.2 m²)
VF = 4.2534tF
0.4934
0
5
10
15
20
25
30
0 10 20 30 40
Filtration time tF [min]
Figure 1: Example of a Typical Filtration Curve
The equation of the trend line for this specific chemical product was calculated by Excel. The
exponent of 0.4934 almost exactly corresponds to the root function and the constant, in this case,
equalled 4.2534. Accordingly, the filtrate flow rate is at its maximum in the beginning and then
declines as a result of the constantly increasing filtration resistance (as the cake thickens).
B. Filtration (continued)
B.1. The important question that follows is what is the optimum filtration time, i.e. when is
the most efficient filtration performance obtained?
The filtration performance (P) is calculated using this formula:
P = VF / (A * ttotal), where ttota l= tF + tside
The side time (tside) includes the washing time, the drying time and the time for filter opening,
cake discharge, closing, cleaning of the filter, if necessary, and other miscellaneous time. Every
batch-operated filter has these side times, which are usually constant from process cycle to
process cycle.
When plotting the filtration performance (P, in liters/m2/hour), for example, and including a side
time of 15 minutes (including washing and drying times), the result is as follows:
Filter performance versus filtration time
0
0,5
1
1,5
2
2,5
3
0 10 20 30 40
Filtration time tF [min]
[l/m²hr]
From the curve, for this example, the maximum filter performance is obtained when the filtration
time of 15 minutes equals the side time of 15 minutes. The mathematical formulas follow on
page 4 of this article.
B.2. Mathematical Proofs for the Filter Performance (P)
Equation (1): P = VF / A * ttotal
Equation (2): ttotal = tF + tside
Equation (3): tF / tF1 = V2 / V1²
By solving equation (1) with equations (2) and (3), the result is
P = (V1² tF / tF1)0. 5 / A (tF + tside)
From Figure 1, (V1² / tF1)0.5 / A = Constant, and therefore, Filter Performance (P) is
P = Constant * (tF
0.5 / (tF + tside))
The condition for the maximum filter performance is based upon the first derivative (d), where
the ratio of derivatives of the Filter Performance (dP) to Filtration Time (dtF) is equal to zero:
Equation (4): dP / dtF = 0
and therefore
Equation (5): (tF
0.5 / (tF + tside))' = 0 (1st derivative = 0)
By solving equation (5), the result is as follows:
0.5 tF
-0.5 (tF + tside) - tF
0.5 / (tF + tside)² = 0
and therefore, the result is
tside = tF
The conclusion is that the side times and the filtration times must be as short as possible and as
equal as possible which leads to thin cake filtration. Of course, one has to ensure that the filter
cake can still be discharged, as a thin cake.
B.3.1. Filtration pressure
The higher the filtration pressure, the higher the filtration performance for a non-compressible
cake. In case of a compressible solid matter, an increase of pressure will often not result in an
increased performance because the filter cake gets more impermeable due to the compression.
Finally, the filtration pressure may also impact the cake-forming layer, which could lead to a
turbid filtrate, if there is a “fine particle tail” in the particle size distribution.
B.3.2. Temperature
The higher the temperature of the slurry, the lower its viscosity. The lower the viscosity, the
higher the filtration performance. As a general rule, therefore, increasing the filtration
temperature will result in an increase in the filtration performance.
B.3.3. Particle Size / Particle Size Distribution (PSD)
The particle size and particle size distribution are important parameters, which can influence the
filtration performance. As the filter cake begins to form, channels (capillaries) are formed
between the particles. The smaller the capillaries, the higher the capillary pressure and thus the
resistance of the filter cake. Small particles and a wide particle size distribution result in a
densely packed filter cake and reduced filtration performance. In cases such as these, a thinner
cake will mitigate this impact and allow for successful filtration. Another alternative would be to
use filter aid, either as a precoat or body feed, to increase the filtration performance.
B.3.4. Particle Shape
The type and shape of the particles also impact the filtration performance. Hard, spherical
particles will form a permeable cake with increased void volume leading to high rates.
Irregularly shaped crystals, platelet-type or flat crystals as well as needles and amorphous
crystals can pack together and result is a dense and low-permeable cake. In cases such as these,
once again, thin-cake filtration, low filtration pressures or filter aid mitigate the impacts of the
particle shape.
C. Washing of the Filter Cake
When washing the filter cake, there are two mechanisms that occur: displacement washing and
diffusional washing. With displacement washing, a large part of the void volume (which is still
filled or saturated with mother liquor before washing starts) is replaced by the washing liquid.
Diffusional washing is the more important mechanism to remove bound or inner liquid. These
liquids are removed by the diffusion washing, i.e. the liquid to be washed out diffuses into the
washing liquid. The driving force is the concentration difference.
C. Washing of the Filter Cake (continued)
Depending upon the cake formation, there may be areas of the cake that have little or no
permeability. These portions of the cake would not be efficiently washed by either mechanism.
In cases such as these, there are generally two alternatives.
The first would be a reslurry wash. The cake would be discharged to a reslurry vessel or
reslurried in-situ. The second alternative, which would require less wash liquid and less
handling of solids, would be a thinner cake to minimize the impermeable areas. This would
allow for a more “plug-flow effect” for the displacement washing. For the bound liquids, there
would be a lower concentration of bound liquids, which would require less diffusional liquids
and less contact time.
Finally, in terms of the parameters that impact washing, these would generally fall in the same
category as the influences on filtration, as described in Section B.
D. Drying of the Filter Cake
When dewatering the filter cake, the liquid remaining in the pores shall be replaced by gas
(mechanical dewatering by blowing) with the objective of obtaining a residual moisture content
as low as possible. During blow-out operation, the capillary pressure of the pores must be
overcome. The smaller the capillaries, the higher the capillary pressure. As soon as the first
capillaries are blown through, the gas consumption rises and reduction of the residual moisture
becomes more ineffective. When this point occurs on the drying curve, there are two further
approaches that are undertaken. First, the cake can be slowly compressed to allow for reduced
gas consumption during this mechanical dewatering stage. Secondly, vacuum drying can be
employed as well as thermal drying.
During the blowing phase, one of the most important factors for a low residual moisture is a
high blow out pressure rather than gas flow. Depending upon the product characteristics, time,
temperature and cake thickness will also impact the drying curve.
Barry A. Perlmutter is currently President and Managing Director of BHS-Filtration Inc., a subsidiary
of BHS-Sonthofen GmbH. BHS is a manufacturer of thin-cake filtration, washing and drying
technologies. Barry has over 25 years of engineering and technical business marketing experience in the
field of solid-liquid separation including filtration and centrifugation and process drying. He has
published and lectured extensively worldwide on the theory and applications for the chemical,
pharmaceutical and energy / environmental industries and has been responsible for introducing and
creating growth for many European companies and technologies into the marketplace. He received a BS
degree in Chemistry (Albany State, (NY) University), MS degree from the School of Engineering,
Washington University, St. Louis and an MBA from the University of Illinois. Barry served on the Board
of Directors of the American Filtration and Separations Society (AFS) and is a member of several
internationally-recognized societies.