Modeling of Filter Cake Washing Performance on Belt and Drum Filters
on Base of Lab Experiments
Jürgen Heuser,
Institut für Mechanische Verfahrenstechnik und Mechanik, Universität Karlsruhe, Germany
Werner Stahl
Institut für Mechanische Verfahrenstechnik und Mechanik, Universität Karlsruhe, Germany
Vom Verfasser zum persönlichen Gebrauch überreicht
Erschienen in „Mineral Processing on the Verge of the 21st Century“
ISBN 9058091724, 1. Auflage 2000
ABSTRACT: An easily usable model for cake washing is derived from experiments. The parameters of the
model can be determined by a minimum of experimental effort. Adopted to the description of the filtration
process on continuous belt and drum filters the model shows that the achievable wash ratio (volumes of wash
water per volumes of pores in the cake) is to a large extend independent of the process parameters frequency
and pressure difference. The influence of these parameters on the final purity of the cake depends on the regime (replacement-transition-time controlled regime) washing ends in.
1 INTRODUCTION
In solid-liquid separation processes one requests
often not only low moisture content of the solid but
also a low content of impurities. In order to reduce
the content of impurity, the filter cake is washed on
the apparatus. Two basic principles of washing can
be distinguished: Replacement washing and washing
by mixing. Subject of this paper is replacement
washing that takes place for example on continuous
belt and drum filters. In that case wash liquor is
pressed as uniformly as possible through the cake by
differential pressure. For design of washing processes various models exist. These models are either
difficult to use, show a lack of accuracy or the multiplicity of parameters in the model is unknown. The
authors of the relevant literature usually give no note
to how the necessary parameters have to be determined.
Here an experimentally supported model is developed from the example of cakes formed from two
quartz sand suspensions (properties in tab. 2). The
experimentally supported methodology has the advantage that it is not necessary to investigate the individual mechanisms (diffusion, desorption, replacement) experimentally. Washing experiments
provide the necessary information. Furthermore it
will be described which experiments are necessary
in order to determine the required parameters with
the smallest necessary effort. Thus, it is possible to
design a wash process on base of a few experiments.
Non-ideal effects as were investigated elsewhere [1]
and such caused by the apparatus are neglected here.
2 ANALYSIS OF WASHING CURVES AND
MODELING STRATEGY
The first step in the experimentally modeling procedure is the analysis of measurements. The model
has to describe the washing curves in their three significantly distinguishable stages: replace-ment, transition and time controlled or diffusion stage. The
measurements are performed on a special nutsch filter, fig. 1. Down to this point the modeling procedure is similar to that presented in [2] and almost
identical to [3]. Deviating from the last authors’
method a mathematical description of the borders
segregating the distinct sections as a function of the
fig. 1: Lab nutsch filter used for experimental work
extrinsic parameters of the process (i.e. in the labexperiment the differential pressure and the cake
thickness) is developed. This step reduces the experimental requirements significantly.
In fig. 2a three washing curves of cakes formed
from a suspension of the solid Quartz SF300 in a
0.1M solution of sodium chloride in demineralized
water are outlined over the wash ratio (eq. 14 in
tab.1). The differential pressure was constant in
these experiments while the cake thickness was different for each curve. In the section of short washing
times the performance of washing follows an ideal
plug flow behavior. The fig. 2b shows the same data
outlined over the washing time. It can be seen that
for long times independent from the cake thickness
all curves run into one straight line. In the transition
stage, i.e. between the first breakthrough of wash
liquor and the time-controlled regime, a straight line
can approximate the performance of washing. In this
regime the gradient of the line depends on the extrinsic parameters differential pressure and cake
thickness.
For experimentally supported modeling the following information has to be provided:
• Breakthrough: point of time tB when the first
time liquor with a content of impurity
smaller than the content in the cake formation phase breaks through the largest capillaries of the cake
• point of time tD when the time-controlled regime is reached.
• correlation between impurity content and
time on one hand and the wash liquor
amount on the other hand - individually for
replacement regime, time controlled regime
and transition section
3 MODELING
3.1 Breakthrough time of the wash liquor and
washing performance in the replacement stage
In the replacement regime the simple plug flow
relation can describe performance of washing:
W
∫c
*
W =0
where t < tB. The definitions for c*, X* and W
can be found in tab. 1. If adsorption is negligible, in
this regime the standardized concentration c* is 1. If
adsorption occurs c* is smaller than 1 but here still
constant. Sorption is negligible here if there is a high
concentration of impurity in the mother liquor or the
particles are coarse with a small specific surface
area. The time of breakthrough tB can easily be detected by measurements of the concentration of impurity in the filtrate: Breakthrough occurs just at that
time when the concentration sinks under the value
measured during cake formation. This point of time
can be calculated from data provided by measurements with the lab-nutsch filter by correction of a
time delay. In fig. 3 such measured times of breakthrough are outlined versus the theoretical time of
breakthrough that is defined as
 ∆p ⋅ PS
V
τ = F = 
VP  ε ⋅ η ⋅ hC2



−1
(2)
The correlation between tB and τ is almost linear.
For small values - due to a negative value for the
point of intersection with the ordinate axis - there is
a deviation from this correlation. In fact the data
seem to be better described by a power law:
(3)
t B = m ⋅τ b
In the example it’s m=0.329 and b=1.18. Due to
the value of b nearby one for large values of τ a linear correlation is a good approximation over a wide
range. For thin cakes the deviation is not acceptable.
The structure of thin cakes seems to differ from that
of thick cakes causing a broader distribution of pore
sizes. This statement is supported from analogous
studies for the much finer quartz SF600 fraction.
While the gradient for SF600 is in a linear approximation higher than for SF300 (0.85 vs. 0.80), the
point of intersection with the ordinate is only half
the value (-2.6 vs. –5.5sec). Both indicates a more
uniform wash flow through the cake and thus a more
linie: regression for all
data w ith W>2
log X* = -0.5*log t/s - 0.47
∆ p=3.5 bar
remaining fraction of impurity X*/-
lines: calculated for
hC 3/6/9/12/17 mm
data points:
hC 6/9/12 mm
0.1
0.01
0.1
1
10
number of w ash replacements W / -
fig. 2a: fraction of impurity vs. number of wash replacements
(1)
(W ) ⋅ dW = 1 − c * ⋅ W
1.00
1
remaining fraction of imputity X* /-
X * = 1−
0.10
12 mm
9 mm
6 mm
0.01
1
10
100
1000
w ashing tim e t W / s
fig. 2b: fraction of impurity vs. time at constant driving force
100
/ 6mm
/ 6mm
/ 6mm
/ 9mm
/ 12mm
/ 15mm
/ 18mm
3.3 Point of entry into the time-controlled stage
50
quartz SF300
in NaCl-solution (0.1 M)
w ash liquor:
demineralized w ater
0
0
50
100
150
τ/s
fig. 3: measured vs. theoretical time of breakthrough
of the wash liquor
uniform pore structure for the finer product. For
cake thickness with practical relevance the assumption of a linear correlation is justified:
(4)
t B = m B ⋅ τ + bB
For small values of bB the wash ratio WB is independent from cake thickness and differential pressure.
With eq. 1 and eq. 3 or 4 washing performance in
the replacement regime for such cases can be described empirically.
3.2 The time controlled stage
For the modeling of the transition stage the
knowledge of the process within the time-controlled
stage will be presupposed. This information will be
provided here. For long washing times in fig. 2b the
performance does not depend significantly on any
extrinsic parameter except the washing time. In this
regime mass transfer limits washing performance.
As the wash result improves very slowly in this regime, it does not matter how fast the preceding replacement stage has been finished. This is true as
long as the driving differential pressure is chosen in
the range usually adapted for filtration. Empirically
time-controlled washing performance can be described by
*
X =X
*
D0
 t
⋅ 
 t D0



−CD
(5)
where CD is the gradient of a regression line for
long washing times in fig. 2b. CD contains mean information about any possible kinetically limited
mechanism (diffusion, desorption, solvation) including mass transfer coefficients and geometrical properties of the pore structure of the cake. tD0 is a reference time necessary to remove the dimension from
the argument of the logarithm. We prefer tD0=1sec.
X*D0 is a hypothetical normalized load of impurity
in the cake at the reference time and can be calculated from regression, too.
The wash ratio at the beginning of the timecontrolled regime can be detected from a more or
less significant break point in each of the curves in
fig. 2a. In less univocal cases the point can be approximated by the point of intersection of the two
lines representing the time controlled regime and the
transition stage.
This point of intersection depends on differential
pressure and cake height. It characterizes the end of
a section where the gradient d[logX*]/d[log(t/s)]
changes significantly. This phenomenon can be understood if the capillary structure of the filtercake is
regarded as a three dimensional network of capillary
segments with a distribution of diameters. Due to
such a structure, statistically the small capillaries
flow into coarse capillaries after a certain mean distance. This mean distance for a homogeneous cake
should not depend on its thickness. With this assumption the time for replacement of the content of
the fine capillaries only depends on the pore velocity
of the fluid. With eq. 14/tab.1 and Hagen Poiseuilles
law it is
tD = te,fine = Li/ufine ∝ hC / ∆p
(6 a,b)
and thus
WD ∝
∆p
1
⋅ tD =
2
hC
hC
In the sense of eq. 6, fig. 4 outlines the wash ratio
WD at the beginning of the time-controlled regime as
a function of the reciprocal cake thickness. WD has
in any case to be smaller or equal to WB. A straight
line with a good correlation can describe the data. In
good approximation one can assume that WD for a
certain cake height reaches WB. That means there is
3
y = 14.26x + 0.1673
2
R = 0.9807
2
WD
tB / s
2,5bar
3,0bar
3,5bar
3,5bar
3,5bar
3,5bar
3,5bar
1
0
0
0.05
0.1
0.15
0.2
-1
(hC / mm)
fig. 4: number of wash displacements at the beginning of the
time controlled stage vs. reciprocal cake thickness
no measurable effect of the distribution of pore sizes
on washing anymore for “thick” cakes, see 3.4.
3.4 The Transition stage
The most complicated stage of washing is the
transition between breakthrough and diffusion regime. In fig. 2a a linear interpolation between the
time of breakthrough and the beginning of the timecontrolled regime seems to be a good approximation. With this assumption kinetics of washing in the
transition stage can be written as:
 W (t ) 

X (t ) = (1 − W B ) ⋅ 
W
 B 
−C B
*
(7)
CB depends on the extrinsic parameters cake
thickness and differential pressure. It can be calculated from the above mentioned information as at
this point calculation of WB, WD, X*B and X*D is
possible.
A transition stage does not necessarily occur in
any washing process: Extrapolation in fig. 4 provides information about a product dependent cake
thickness that causes replacement to flow directly
into the time-controlled regime. The critical cake
thickness in our example has roughly measured to be
in the order of magnitude of 1700 particulate diameters. In the case of other investigated particle systems we could not find a transition stage for – like
Quartz SF600, Silicic acid, and a Dye – we did
never fall below that number of theoretical cake layers. Data taken from [4] for a blue dye do not show a
transition stage, too. Evaluation of those data gave a
ratio of hC/x50,3 of 5000 but that of course is above
the critical number of layers.
Judged from the view of the wash liquor consumption the most efficient way to reach a purity at
the starting point of the time-controlled regime is to
adopt the critical cake thickness. The remaining impurity at the time of breakthrough can only be removed by accepting the time limitation - presupposed one wants not to destroy the cake. At the same
time it makes sense to reduce differential pressure,
pause in washing process or more rigorous reslurry
or provide a change of the pore structure. The decision for or against such means depends on economical criterions and cannot be discussed in detail here.
4 EXPERIMENTAL DESIGN
Of course in the example modeling was easy due
to a broad base of experimental data. For a unknown
product the extent of the experimental procedure has
to be limited by the following strategy:
• Systematic variation of the wash ratio W in the
time controlled regime: Wash experiments for
three combinations of cake thickness and pressure
difference at variable pore velocities, i.e. three
different ratios of ∆p/hC and washing times. The
cake height should be varied quite below the expected critical ratio of hC/x50,3 of ca. 1000. Measured data: Concentration of impurity in the filtrate
as a function of time, content of impurity in the
cake after washing, cake porosity and permeability; calculated parameters: exponent CD and impurity load X*D0 at reference time tD0 from regression with eq. 5 and fig. 2b; time of
breakthrough from online measurements of concentration in the filtrate for each experiment
tB,i(∆p,hC) for fig. 3; by means of linear regression parameters mB and bB in eq. 4 from fig. 3.
• Wash experiments with a wash ratio surely in the
transition stage, approximately W=1.25: Variation of ∆p/hC over at least three steps. Linear extrapolation in fig. 2a starting at WB through the
measured point down to the point of intersection
with the line that describes the time controlled regime for each chosen ∆p/hC (in double logarithmic scale). This provides WD,i, and with eq. 14
and 16 tD,i. Empirical parameters in eq. 5 from
fig. 4 for WD,i outlined as a function of 1/hC by
linear regression.
5 FORMULATION FOR CONTINUOUS BELT
AND DRUM FILTERS
The connection between lab experiment and the
rotary filter process is given by the correlation between section time ti, section area in relation to the
complete area αi and frequency n (i.e. revolutions
per second) of drum or belt:
ti =
L
αi
with α i = i for belt
n
Lbelt
and α i =
(8)
ϕi
for drum filters
360°
The main question that will be answered here is
which parameters have an influence on the wash ratio and the level of purity of the cake that can be
reached on a continuous filter.
5.1 Sensitivity of the wash ratio for changes in
process parameters
With eq. 8, eq. 16 (Darcys law), and eq. 15 for
description of the cake formation for an incompressible cake the wash ratio W in eq. 14 can be
written as
W=
αW
1
⋅
⋅
α1 2 ⋅ ε ⋅ κ
1
h
1 + CE
hC
(9)
5.2 Performance in the replacement stage
Eq. 1 under assumption of a negligible influence
of adsorption gives
with W ≤ WB
(10)
W can be calculated from eq. 9. Thus, in this
stage the sensitivity of X* for the individual parameters is the same as discussed for the wash ratio W:
There is no influence of the frequency n or differential pressure ∆p on impurity content.
For what range of W is this valid? If the relation
between tB and τ is a straight line through the origin,
the wash ratio WB does not depend on cake thickness and differential pressure and of course also not
on n. In a more general case one gets from eq. 4
WB = m B +
bB
τ
(11)
Thus, as long as the ratio bB/τ is small in comparison to mB the wash ratio WB is nearly independent from τ and thus from ∆p and n, too.
5.3 Performance in the transition stage
Transition stage is fixed with the two conditions
W > WB and n ≥
αW
tD
Replacing time in eq. 5 by eq. 8 results in
*
where hCE = PC ⋅ R M . As can be seen under the
chosen assumptions the approachable wash ratio W
is only slightly dependent from differential pressure
∆p and frequency n. If the cake resistance is much
higher than the cloth resistance the last term in eq. 9
is negligible and W does not depend on these parameters at all. No parameter adjusted to the apparatus except of the chosen geometric ratio αW/α1 has
an influence on the wash ratio. All other parameters
in eq. 9 can usually in process not be changed systematically. Thus for a given apparatus the wash ratio is a result of the chosen geometry.
This inconvenience can be overcome by constructional means e.g. by providing that differential pressures in cake formation section and in the washing
section are independent from each other.
X * = 1−W
5.4 Performance in the time controlled stage
(12)
where tD can be calculated with WD from eq. 14
and eq. 16. Eq. 5 can be used to calculate X* with W
from eq. 9.
X =X
*
D0
 α
⋅  W
 n ⋅ t D0



−CD
with n <
αW
tD
(13)
This equation shows an influence of frequency n
on X*. An influence of differential pressure – under
the same assumptions as discussed for lab experiments modeling – does not exist in this equation.
6 CONCLUSIONS
Performance of washing can but has not in any
case to show three different stages: The second stage
can be eliminated by a properly chosen cake thickness. For the investigated products the critical cake
thickness seems approximately to be in the order of
magnitude of 1000-2000 particle diameters (x50,3).
A look at the derived equations for the performance of filter cake washing on belt- and drum filters
shows interesting tendencies. The approachable
wash ratio on such filters depends for incompressible products only slightly on the controllable process
parameters frequency n and differential pressure ∆p.
Only a constructional separation of the adjustment of
pressure difference in cake formation and washing
section gives a degree of freedom for controlling
wash ratio. Thus, the ratio αW/α1 is crucial to a large
extent for the washing stage that can be reached on
such a filter.
Thus, washing performance in replacement stage
is not depending on differential pressure or frequency whereas transition and time controlled regime can be influenced by frequency.
The model of course describes the washing behavior of the product under ideal conditions only.
The apparatus can influence this behavior negatively. Thus, pilot scale tests are still necessary.
7 ACKNOWLEDGMENTS
The authors gratefully thank the German
AiF/GVT and the Bundesministerium für Wirtschaft
(BMWi) for supporting this work. The work was
supported by funds of the BMWi
V f
8 APPENDIX
8.1 Basic definitions and equations
W=
VW
VP
(14)
2
hC = a k ⋅ hCE
2 ⋅ PC ⋅ κ ⋅ t1 ⋅ ∆p
+
− hCE
ηl
hCE = PC ⋅ R M
[5]
with
(15)
PC ⋅ ∆p
V
=
⋅t
A η ⋅ (hC + hCE )
X* =
c* =
(16)
X (t )
ε
with X (t = 0) = X ads + c f ,0 ⋅
X (t = 0)
(1 − ε ) ⋅ ρ S
c f (t )
c0
(1 − ε ) ρ s
Vs
with c 0 = X(0) ⋅
= X (0 ) ⋅
⋅
ε
ρl
VP
W
WB
m³/s
-
WD
X
kg/kg
X*
X*D0
Xads
kg/kg
xz,3
∆p
α1
m
Pa
-
ε
ϕi
κ
ρ
τ
°
kg/m³
s
(17)
REFERENCES
(18)
[1]
Heuser, J.
Stahl, W.
[2]
Wakeman, R.J.
[3]
Bender, W.
[4]
Bender, W.
[5]
Anlauf, H.
8.2 Particle diameters of the discussed products
X10,3
2,5 µm
0,8 µm
quartz SF300
quartz SF600
X50,3
8 µm
3 µm
X90,3
20 µm
8 µm
NOTATION
αW
bB
c*
CD
cf,0
s
g/L
hC
hCE
m
m
Lbelt
Lfine
Li
mB
n
PC
PS
m
m
m
1/s
m²
m²
ri
t1
tB
tD
m
s
s
s
tD0
te,fine
tW
u
ufine
ui
V
s
s
s
m/s
m/s
m/s
m³
flux of filtrate
volumetric wash ratio
wash ratio W at time of breakthrough
W at entry in time-contr. Stage
total mass of impurity per mass
solid
remaining fraction of impurity
fraction of impurity at tD0
mass of adsorbed impurity per
mass solid
particle size z percent are smaller
differential pressure
ratio cake formation area to total area
porosity
angle of section i
concentration parameter
density, index S=solid, l = liquid
theoretical time of breakthrough
ratio washing area to total area
parameter breakthrough
reduced concentration
parameter time controlled stage
filtrate conc. at starting point of
washing
cake thickness
equivalent cake thickness of the
filter cloth
length of the filter belt
mean length of a fine capillary
length of the belt in section i
gradient in eq. 4
frequency
permeability of cake
permeability of cake inclusive
cloth
radius of the class i
cake formation time
time of breakthrough
time to entry in time controlled
stage
reference time
time to empty fine capillaries
washing time
velocity
mean velocity in fine capillaries
velocity in class i
Volume; index P for pores, l liquid
The Influence of Non-Ideal Effects on Cake Washing Advances in Filtration and Separation Technology 12 (1998),
555-561
Transport equations for filter
cake washing, Chem. Eng.
Res. Des., Vol. 64, July 1986,
pp. 308-318
Das Auswaschen von Filterkuchen, Chem.-Ing.-Tech. 55,
Nr. 11, Verlag Chemie
GmbH, Weinheim Germany
(1983)
Filtrieren, Auswaschen und Entfeuchten feindisperser Feststoffe,
Chem.-Ing.-Tech.
48(1976)/Nr. 4
Entfeuchtung von Filterkuchen
bei der Vakuum- Druck- und
Druck-/Vakuumfiltration
Dissertation
Universität
Karlsruhe, VDI Verlag, Fortschritt-Berichte, Reihe 3, Nr.
114, 1986