Vom Verfasser zum persönlichen Gebrauch überreicht
Erschienen in
Advances in Filtration and Separation Technology 15 (2001) (CD-ROM)
INVESTIGATIONS ON WASHING OF COMPRESSIBLE FILTER CAKES
Dipl.-Ing. Jürgen Heuser*
Dipl.-Ing. B. Hoffner
Dr.-Ing. H. Anlauf
Prof. Dr.-Ing. Werner Stahl
juergen.heuser@ciw.uni-karlsruhe.de
bernhard.hoffner@ciw.uni-karlsruhe.de
harald.anlauf@ciw.uni-karlsruhe.de
werner.stahl@ciw.uni-karlsruhe.de
Institut für Mechanische
Verfahrenstechnik und Mechanik
Universität Karlsruhe (TH)
D-76128 Karlsruhe, Germany
Phone: +49-721-608-2401
Fax: +49-721-608-2403
Abstract: In separation theory highly disperse particle systems are subsumed under the class
of the compressible products: They form cakes with high porosity and structural parameters
depending on the scale of the forces applied to the cake or sediment. We focus herein on the
special characteristics caused by the surface effects. The class of impurities we look at are
soluble salts.
Equilibrium in dewatering by differential pressure or centrifugation can easily be measured as
capillary pressure curve or Bond diagram assuming or forcing negligible influence of thermal
effects. As the latter cannot be neglected for cake washing, such equilibrium has to be defined
under certain assumptions. We define it herein as equilibrium inside of thin layers of the cake,
i.e. the hypothetic case of completely adjusted equilibrium of diffusion and sorption at any time
in any differential plane located vertically to the wash flow. One can get very near to this
hypothetical case by slowing the wash flux down. We assume for the definition of equilibrium
the short distance relaxation of concentration gradients would dominate in relation to the longrange diffusion, which is orientated up-stream opposing to the axial flow of wash liquor. Axial
dispersion by hydrodynamic effects is neglected, too.
Experimentally determined information of typically non-linear sorption isotherms was used, the
experimental procedure for their determination is discussed in detail.
By means of simulation, the optimal achievable purity of a cake in displacement washing is
described for a variety of parameters and typical cases of process design. Included is counter
current washing in mixing-settling cascades and on belt filters. For certain cases simulation is
compared with experimental washing results from the lab. It is proven that equilibrium of
sorption cannot just influence but even govern the washing process for highly disperse systems
at very high purity.
INTRODUCTION
The dynamic development of the nano-technologies indicates the increasing economical
importance of purification processes for highly disperse particle systems. Examples for such
products are highly disperse silica or dyes. In life science technologies highly disperse silica
gets increasingly common as carrier for active substances. Specific surfaces up to 400 m² / g
are not seldom in the latter, whereas typical mineral products in the micron scale of particle size
seldom reach more than 10 m²/g. In many cases the purification process is realized by washing
a cake formed from aggregates of the nano-scale particles on a filter. Surface effects including
sorption can affect these processes. The challenge of washing such products is the inherent
discrepancy between the demand for a very high purity on one hand and on the other hand the
fact that sorption increases the effort exponentially with this demanded purity. An optimum
design procedure for such a washing process has to take care for the equilibrium limiting a
certain washing method.
Investigations of counter current multi staged washing in the lab are quite sophisticated.
Simulation of kinetics and information about the optimum reachable result by improving the
parameters of operation on continuous filters (belt, drum etc.) would be helpful to minimize the
experimental effort.
In [1] it has been shown how one can track the shape of single staged washing curves by
means of a simple experimentally based model. The suggested model was developed under
assumptions like constant quality of washing liquor and one staged washing. The main
advantage of that kind of description was that it provided a strategy for the reduction of the
experimental effort. As a substantial disadvantage such empirical models cannot describe
cases with changing boundary conditions (i.e. quality of wash liquor, porosity of the cake).
Thus, for more sophisticated process designs like counter current washing a differential
approach will be chosen here.
IDEA OF THE EQUILIBRIUM MODEL
We define the equilibrium of washing as a case of completely adjusted equilibrium of diffusion
and sorption at any time in any differential plane located vertically to the flux of wash liquor.
Thus, concentration gradients out of porous particles and interstices are assumed to be
completely equilibrated. This does not mean they are no more existing: Sorption at the surface
and inside of the double layer around of the particles causes gradients that are kept constant by
electrostatic and van der Waals forces [2]. The flux of wash liquor is assumed to be plug-like.
Thus, we assume the short distance relaxation of concentration gradients dominate in relation
to the long range diffusion which is orientated in opposite to the axial flow of wash liquor. For a
washing process these assumptions are theoretically valid at low Peclet numbers Pe = u * x50,3 /
D combined with high Bodenstein-numbers Bo= u * hC / D.
MEASUREMENT AND MATHEMATICAL DESCRIPTION OF SORPTION ISOTHERMES
For comparison and calculation we need two types of experiments. One type is the standard
washing experiment on a nutsch filter. A description of the experimental set-up, a strategy to
reduce the experimental effort and typical washing diagrams can be found in [1]. Furthermore
for the description of cases dominated by adsorption one has to perform experiments to
determine the sorption equilibrium between liquid and solid phase.
For that means a couple of suspensions with well defined masses of solids, de-ionized water
and variable salt concentrations were equilibrated. We selected two different salts as impurities
(NaCl, Na2SO4) and the insoluble solids TiO2 and quartz (99% SiO2) for our investigations. As
liquor we used de-ionized water (conductivity <2µS/cm). The solids volume concentration cv had
to be chosen depending on the type of solid. For quartz we chose cv=20%, whilst the high
disperse TiO2 made it necessary to work with cv=10%. The concentration of the clear
supernatant liquid was determined after a time of more than 1 day for equilibration and
sedimentation at a temperature of
tab. 1: properties of the systems of matter
about 20°C. Analysis had to be done
Sm
solid
x
ρ
salt
k
n
by conductometry. As solids are not
quartz SF600
3
4.4
2.7
NaCl
3.94E-06
0.6353
quartz SF600
“
“
“
Na SO
1.29E-05
0.6974
ever absolutely pure, an excess
TiO
0.75
15.9
4.0
Na SO
6.67E-05
0.6560
µm
m²/g
g/cm³
mol/g
conductivity in the suspension exists.
Ignoring this fact can cause
TiO2 / Na2SO4
1.E-04
tremendous errors especially for
SF600 / Na2SO4
measurements at low concenSF600 / NaCl
Freundlich isotherme
trations. As there was no information
1.E-05
available about the ionic species
causing this base conductivity, the
1.E-06
equivalent-concentration
of
an
equilibrated
“zero-probe”, i.e. a
1.E-07
probe made from pure water and
solid without any addition of salt
under identical conditions, was
1.E-08
0.0001
0.0010
0.0100
0.1000
1.0000
subtracted from the measured
[mol/L]
bulk concentration c
values:
fig.1: Sorption isotherms of the three systems of
K
o
zero
matter investigated. Freundlich-type due to linearity
c eq = c eq κ eq
− c eq
(1)
in double logarithmic scale
This equation is valid only for small
concentrations in the zero probes: The slight non-linearity of the relationship between
conductivity and concentration would cause an error. Furthermore we assume there is no
adsorption of this base impurity in concurrence with our model impurities. This for sure is not
right and causes some deviations in our measured equilibrium at small concentrations. These
deviations are reduced by evaluation with linear regression equivalent to fig. 1 - provided the
type of isotherm chosen is correct.
50,3
solid
4
2
4
F
specific adsorbed amount q / (mol/g)
2
F
2
GGW
{ }
By a mass balance the load of salt q that has been eliminated by sorption
K
q = c eq ⋅ Vl / M S
(2)
can be calculated. Fig. 1 shows the isotherms we have determined for the regarded systems.
For quartz SF600 the adsorbed amounts of salt (NaCl, Na2SO4) is very small compared to the
case of TiO2/Na2SO4. The tendency to adsorb ions from a solution increases dramatically with
the specific surface of the solid and the charge of the ions.
For mathematical description of the sorption isotherms a number of theoretical approaches
exists. Very common are the Langmuir-type and the Freundlich-type. The Langmuir-type in
opposite to the Freundlich-type is characterized by a limit for the adsorbed mass. As the
isotherms in fig. 1 do not converge, we decided to choose a Freundlich-type:
q=k
⋅  c / c° 
F  eq

n
(3)
This decision is proven to be right: In double logarithmic scale each of the three isotherms
results in a straight line, fig. 1. The slope provides the Freundlich-exponent n, the value of q at
ceq=c° is the Freundlich-coefficient kF. A collection of the parameters for the investigated
systems and further information can be taken from tab 1.
CONSEQUENCES OF SORPTION FOR THE DISTRIBUTION OF IMPURITY BETWEEN
SOLID AND LIQUID PHASE
The importance of adsorption in the washing process especially at high purities can be
estimated by fig. 2. Distribution coefficients KD (=adsorbed mass/total mass in the system) of
the salt between the solid and liquid phase as a function of the solid concentration on one hand
and of the salt concentration on the other hand are shown there. The graph is true for a
homogeneous suspension. The investigated system was TiO2/Na2SO4. Starting for example at
a chosen cv=10% / c=0.1 mol/L gives KD≈ 5%. Moving to cv=40% - that’s equivalent to the
concentration in a filter cake of this
product – KD is increased up to ≈25%.
0,7
TiO / Na SO (aqueous)
ceq= 1 mmol/l
This is true under the assumption
0,6
ceq= 10 mmol/l
sorption would behave at any time
ceq= 100 mmol/l
0,5
equivalent to the isotherm, fig. 1.
0,4
Possible dependencies from the solids
washing
concentrations are neglected. The
0,3
vertical arrow in fig. 2 illustrates the
0,2
direction of change of the distribution
0,1
cake formation
ratio during cake washing: The fraction
0,0
adsorbed on the surface of the solids
0,0
0,1
0,2
0,3
0,4
0,5
inside of a layer of cake increases and
vol.-concentration of solids c [-]
reaches KD≈70% for layers with a bulk
concentration of 1mmol/L. Thus, the
achievable purification ratio x* that can
fig. 2: Adsorbed fraction of impurity in a
be reached by displacement decreases
suspension depending on solids and bulk
dramatically with growing initial purity.
concentration
2
4
ads. fraction of impurity K D [-]
2
V
EQUILIBRIUM MODEL FOR MIXER-SETTLER-DEVICES
An alternative to cake washing and a very common procedure is mixing in a stirred tank and
separation by methods like settling, centrifugation or filtration afterwards. In fig. 3 an elementary
cell representing such a procedure and the relevant fluxes of matter are illustrated. Very often
these elementary cells are combined in cascades with counter current flux of liquor and solids.
An analogous application is the paddle washer [3].
We assume constant solids concentration at the in- and outlet of each cell, stationarity, ideal
mixing and spontaneous adsorption inside of the mixing tanks here. Thus, the mass balance for
the concentration of impurity inside of the kth cell is (fig. 3):
M S ⋅ (q k −1 − q k + F ⋅ (c k −1 − c k )) + Vw (c k +1 − c k ) = dN / dt = 0
(4)
with Vw = W ⋅ ε ⋅ Vc = W ⋅ F ⋅ M s . Eq. 4 cannot be solved analytically. We chose the strategy to
Vw
Ck+ 1
M s
M s
qk −1
qk
F
ck −1
F
ck
Vw
ck
remaining fraction of impurity X* [-]
estimate the concentration in the last elementary cell and compute up-stream. Thus the
concentration in the bulk of the inlet of the first stage is obtained. As long as this calculated inlet
concentration has an intolerable deviation from the known true one the calculation is repeated
with a new estimation of the concentration in the last cell.
1.0E+00
adsorption
1.0E-01
1.0E-02
TiO2/ 0.1M Na2SO4
cw=0 mol/L
F =3,73e-4 L/g
W=4
1.0E-03
1.0E-04
1.0E-05
neglecting
adsorption
1.0E-06
1.0E-07
1
2
3
4
5
6
7
8
9
10
number of mixing-settling cells nk
fig. 3: Elementary cell for counter
current mixing-settling
fig. 4: Comparison of nk-staged mixing –
settling with and without sorption
In fig. 4 the purities achievable by such a multi staged mixer settler at a certain wash ratio have
been calculated with and without sorption. The system was TiO2/0.1 M Na2SO4. The effect of a
higher number of stages is dramatically overestimated for purities beyond an absolute load X* =
1e-3 in the latter case. This emphasizes again the importance of keeping adsorption in mind for
this type of products.
MODELLING OF THE CAKE WASHING EQUILIBRIUM
As described we regard the equilibrium of cake washing to be governed by sorption in discrete
thin layers of the cake. As the description of counter current washing on a belt- or drum filter
follows later on, we choose a numerical method to calculate the washing progress.
We equidistantly divide the cake into nx layers, the total washing time into nt steps. Furthermore
we assume that the concentration in the bulk liquid in each of the layers is constant. During a
time step, the liquid content of each layer moves down one layer-position towards the filter
cloth. The upper layer is filled with pure wash liquor. Sorption equilibrates in each layer
spontaneously at the end of the time step. Mathematically this methodology can be described
by
q{i, j} ⋅ Vs ⋅ ρ s + Vl ⋅ c {i, j - 1} = q{i + 1, j} ⋅ Vs ⋅ ρ s + Vl ⋅ c{i + 1, j}
(5)
with t = i ⋅ ∆t , x = j ⋅ ∆x , ∆t = tW / nt , ∆x = hk n x
(5b)
Eq. 5 rewritten with eq. 2 is
 c{i, j} 
 c{i + 1, j} 
kF ⋅
 Vs ⋅ ρ s + Vl ⋅ c{i, j - 1} = k F ⋅ 
 ⋅ Vs ⋅ ρ s + Vl ⋅ c{i + 1, j}
c°
 c° 


n
n
(6)
Where in general 0<n<1. Eq. 6 can – with some exceptions – not be solved analytically. A
numerical method has to be used.
The accuracy of such a scheme can depend strongly on the numerical parameters. fig. 5
compares calculations for the case of a single staged washing of quartz SF600. In the range of
the numerical parameters selected for the three cases, there obviously is an influence of nt and
nx on the result of calculation. Comparing the deviation to the accuracy of measurements, it is
tolerable to use even the low number of layers and time steps.
The case of multi staged washing on a continuous belt- or drum filter can now be described
easily and quite analogous to the single staged case, fig. 6: A compartment of cake is traced on
its way through the nz washing stages. On each of these stages it is washed with wash liquor of
a certain concentration of impurity. This concentration of impurity is unknown except for the last
stage. We assume the effluent filtrate on each washing stage k to be mixed ideally before it is
used on the stage k-1 as wash liquor. Thus, the quality of wash liquor is constant on each stage
remaining fraction of impurity X*= N/N0
1
number of time steps x number of slices /
computing-time (Intel PIII500):
181 x 45: 1s
901 x 227: 25 s
3601 x 908: 390 s
Cex(2) Cex(3)
∆V
∆V
0.1
CW,
∆V
...
TiO2 / 0.1 M Na2SO4 (aqueous)
Cex (1)
0.01
0
1
2
3
4
5
wash ratio W
fig. 5: Influence of the chosen numerical parameters on
the calculated washing result
Cex (2)
Cex(nz)
∆V
∆V
∆V
fig. 6: Scheme of a staged filter
in the stationary case. As the purity of the effluent on stage k influences itself - it is lead back to
stage k-1 - a mathematical solution can only be found by iteration: The concentrations of the
wash liquor on all stages except of the last stage nz have to be estimated in a first step. With
these estimated concentrations the effluent concentrations for each of the washing stages can
be calculated and compared to the estimated. As long there is intolerable deviation of estimated
and calculated concentrations of the wash liquor on each stage, the calculation is repeated with
a new estimation. Convergence of this iteration scheme occurred to be quite good. The iteration
can be terminated when the calculated effluent on each stage k is with some tolerable deviation
identical with the estimated impurity of the wash liquor.
MODEL REGARDING KINETICS
The suggested scheme can be modified with regard to additional mechanisms. The assumed
new structure of a layer of cake can be seen in fig. 7: We separate a stagnant region I from a
flow channel II as suggested in [4]. Both regions again are
separated in liquid and solid phase. Mechanisms that shall be
regarded here are horizontal diffusion from stagnant regions
Drad
into the flow channel, axial diffusion (or dispersion) up-stream.
To ease the mathematical handling of the model, we assume a
quasi-stable process in any time step ∆t, ideal mixing inside of
each of the two regions and constant resistance for transport of
mass. Again convective transport is decoupled from the other
mechanisms by assuming the liquid volume in section II of
I D
DkonvII
ax
each layer moves one layer downwards and all other
fig. 7: Separation of the mechanisms take place spontaneously at the end of each time
matter in stagnant and step. The mass balance for the impurity in each of the regions
convected regions
at a certain time step i in a layer j can now be written as
VsII ⋅ ρ s ⋅ (q{i, j} − q{i + 1, j}) + VLII ⋅ (c {i, j - 1} − c{i + 1, j}) = ∆t (Drad + Dax )
VsI ⋅ ρ s ⋅ (q{i, j} − q{i + 1, j}) + VLI ⋅ (c {i, j} − c{i + 1, j}) = ∆t (− D rad )
(7a,b)
With the assumption of short time steps ∆t and a gradient of concentration that is only slightly
changing in a time step ∆t we describe the two fluxes of diffusion in eq. 7 as:
(
)
D ax = ∆NII ∆t = −k ax ⋅ (c II {i − 1, j + 1} + c II {i − 1, j − 1} − 2 ⋅ c{i − 1, j})
Drad = ∆NI ∆t = −k rad ⋅ c II {i − 1, j} − c I {i − 1, j}
(8a,b)
Thus, the transported mass in every time step can be calculated by means of eq. 7, 8 with ∆t
and Darcy’s law – provided it is allowed to use the latter.
EQUILIBRIUM OF CAKE WASHING: VALIDATION FOR SINGLE STAGED CASES
Simulation of the washing process with the equilibrium model unveils a strong dependency of
the success of washing from the initial impurity level, fig. 8. Thus it is not allowed to assume a
constant ratio of reduction of the impurity level for each stage of a counter-current washing
process or after an intermediate mixing and settling.
Comparison of the experimentally determined washing results with simulated curves shows a
large deviation in the case of quartz SF600/0.1 M NaCl, fig. 9. For TiO2/0.1 M Na2SO4 there is
only a slight deviation. The deviation is a measure for the lack of ideality. The non-ideality is to
a large extent caused by differing time-scales of diffusive reduction velocity and convective flux.
The potential to improve washing in the case of TiO2 by synchronization of convective flux and
mass transport by diffusion is small. Washing SF600 is obviously governed by slow diffusion: It
shows a large potential for reducing the amount of wash water needed for a certain level of
purity by making a break for diffusion or reducing the wash liquor flux.
TiO2 in Na2SO4 solution (aqueous)
1.0E+00
1.0E-01
remaining fraction of impurity X*= N/N0
remaining fraction of impurity X*= N/N0
1.0E+00
1.0E-02
1.0E-03
1.0E-04
1.0E-05
0.001 M
0.01M
0.1M
1M
1.0E-06
0
0.5
1
1.5
2
2.5
wash ratio W [-]
fig.
8:
Influence
of
the
initial
concentration on the shape of the
washing curve for cases influenced by
sorption (simulation)
1.0E-01
1.0E-02
1.0E-03
TiO2 in 0.1M
Na2SO4 sol.
1.0E-04
SF600 in 0.1M
Na2SO4 sol.
1.0E-05
SF600 in 0.1 M
Na2SO4 sol.
1.0E-06
0
0.5
1
1.5
2
2.5
wash ratio W [-]
fig. 9: Simulated equilibrium and
measured washing results in comparison
Comparison of fig. 8 with the multi staged mixing-settling in fig. 4 shows a great advantage for
single staged cake-washing: The equilibrium of single staged filter cake washing at W=2 can be
estimated in fig. 9 to be X*=5%. One needs a 4 staged mixing-settling at W=4 to reach the
same result, fig. 4.
MULTI STAGED EQUILIBRIUM OF WASHING
Multi staged washing procedures are frequently used to have a maximum concentration
difference for diffusion and adsorption at the end of the washing process. In fig. 10 a case
governed by sorption (TiO2/1e-3 M Na2SO4) was simulated. Compared are a 3-staged counter
current and a single staged simulation. The wash ratio was the same in both cases.
There is only a small advantage of the multi-staged application. On the second stage there is
only a slight decrease of impurity. In the shape of the curves a displacement phase can only be
detected for the first stage. The second and third stages show a monotone change: At the high
purities the largest part of the remaining impurity is no more replaceable as it is adsorbed at the
solids surface.
remaining fraction of impurity X*=N/N0 [mol/mol]
single stage
quasi-continuous: τ = t / tcycle
continous: τ= x / Lstage
1
1st of 3 stages
2nd of 3 stages
3rd of 3 stages
W=4
0,1
stage 1
stage 2
stage 3
0,01
0
0,5
1
1,5
2
2,5
3
τ / [-]
fig. 10: Comparison of simulated 3-staged washing and single
staged washing under equilibrium conditions for W=4. Only
slightly deviating final result
MULTI STAGED WASHING REGARDING KINETICS
remaining fraction of impurity X*= N/N0
The case of kinetically influenced
washing shall be shown for a coarse
product with only slight tendency to
1.00
Quarz SF300/0.1M NaCl (aqueous)
adsorb impurities.
From fig. 11 the volume of the stagnant
region in the filter cake and the mass
transfer coefficient were estimated for
the kinetically dominated washing of
0.10
quartz SF300/NaCl (x50,3=10µm, 99%
SiO2). The validity of the single staged
washing curves is acceptable up to
approximately W=5. With an estimated
stagnant pore volume of 4% of the total
0.01
cake volume and mass transfer
0.1
1.0
10.0
coefficients krad=2e-11 m³/s, kax=0 (ref. to
wash ratio W [-]
eq. 8) we obtain the washing curve for 3
fig. 11: Adaptation of stagnant volume and mass stages in fig. 12.
transfer coefficient for quartz SF300: Calculated The shape of washing curve on each of
curves and measured data
the stages shows an unsteady change
where the washing front breaks through
the cake. Obviously on each stage a
N/N 0
displacement and a diffusion phase exist. For comparison two single staged simulations have
been transformed: The first is for washing on the area of just one of the three stages with the
same pressure difference and thus the same amount of wash liquor. The second curve was
calculated assuming that the
area of all three stages can be
Quarz S F300/0.1M NaC l (aquous),
1,0E +00
used for single staged washing
W =2, ∆ p=3.5 bar, hc =9mm
with the same total amount of
fresh wash liquor as employed
in the multi-staged case. Thus,
1,0E-01
the pressure difference in that
case is assumed to be reduced
to a third of the proceeding
stage 1
case.
stage 2
Multi staged washing gives a
1,0E-02
much better result than any of
these two single staged cases.
stage 3
quasi-continuous: τ = t / tcy cle
continous: τ = x / L stage
CONCLUSIONS
1,0E-03
0
0,5
1
1,5
2
2,5
3
Sorption effects are proven to
have a large importance for
washing. They govern the
progress of washing for nonfig. 12: kinetically influenced washing, counter current
linear sorption isotherms, high
multistage; system quartz SF300 / 0.1M NaCl
specific surfaces and high purity
demands or/and low initial
impurity contents. By means of simulation with the suggested equilibrium model one can
appreciate very easily, whether a certain need for purity can be reached or not. Deviation of the
calculated equilibrium from the measured results of typical washing experiments is a measure
for the importance of kinetics of mass transport for the special washing process. A small
deviation is a hint for a sorption-governed process. For such a system the question is how to
get as much as possible washing liquor in contact with the solid. For a process with a large
deviation the question is how to reduce non-idealities and how to shorten the distance the
impurity has to overcome by means of diffusion.
Multi staged washing on filters is not of much use for sorption controlled cases whereas for
diffusion-controlled cases it can be favorized.
τ [-]
1st of 3 stages
3rd of 3 stages
single-staged, 1.2bar
2nd of 3 stages
single-staged 3.5 bar
ACKNOWLEDGEMENTS
This work was supported by funds of the “Deutsche Forschungsgemeinschaft” (DFG,
Geschäftszeichen AN248/4-1). We gratefully thank the DFG for this support.
LITERATURE
[1] Heuser, J.; Stahl, W.:
Experimentally Supported Modeling of Filter Cake Washing Performance, Advances in Filtration and
Separation Technology 14 (2000), 447-454
[2] Hunter, R. J.:
Zeta-Potential in Colloid Science, Academic Press, London 1981
[3] Stahl, W.; Langeloh, T.:
The paddle washer – a mighty clean washer; Aufbereitungstechnik 6 (1988)
[4] Wakeman, R.J.; Rushton, A.:
A structural model for filter cake washing; Chem. Eng. Sci. 29 (1974)
NOTATION
Bo
c
c°
mol/L
mol/L
Bodenstein number
concentration of the impurity
normal concentration =1 mol/L
ceq
mol/L
mol/L
corrected concentration in equilibrium
concentration in equilibrated state
mol/L
mol/L
m²/s
m²/s
Pa
m
m
L/g
m
-
apparent concentration in equilibrated state due base conductivity
concentration in the filtrate
volume concentration of solid
diffusion coefficient
flux of matter by axial diffusion or dispersion, radial diffusion and convection
driving pressure difference
duration of a single time step
thickness of a layer
porosity
ratio volume liquid per mass solid in cake or sediment
cake height
index of the time step, the layer of cake, the stage or elementary cell
S/m
m³/s
mol/L
m
conductivity in equilibrium state referred to 25°C
axial and radial mass transport coefficient (m³/s due to unknown area)
distribution coefficient of the impurity
Freundlich coefficient
total length
kg/s
kg
mol
mol/g
mass flux of solid
mass of solid
amount of matter, initial amount of matter
Freundlich exponent
total number of elementary cells, time steps, cake layers, washing stages
Peclet number of the particle
adsorbed amount of matter per mass of solid
c eq , cggw
zero
c eq
cex
cV
D
Dax,Drad,Dconv
∆p
∆t
∆x
ε
F
hC
i, j, k
o
κ eq
kax, krad
KD
kF
L
M S
MS
N, N0
n, nF
nk,nt,nx,nz
Pe
q
ρS
Sm
t,tcycle
τ
u
kg/m³
m²/kg
s
m/s
solids density
mass specific surface of the solid matter
time, time of the cycle
time related to the total cycle time or position in a zone related to the total length
velocity of the wash water in the cake
zone
V phase
m³
volume of phase (liquid (l) or solid (s)) in the balance zone I or II
m³/s
m
m
volume flux of cake or sediment, wash liquor flux
volume wash liquor per pore volume of the cake or sediment
absolute position inside of a stage or the cake
impurity amount remaining related to the amount at the beginning
mean particle size
VC , Vw
W
x
X*
x50,3