6.

3 Formation of Incompressible Filter Cakes by Pressure Filtration 95

Figure 6.10 Principal situation during cake formation. p1

hc

p2
p0 heq

all practically relevant parameters. For this purpose, some boundary conditions
have to be defined, which can be realized not only in theory but also in practice.
The slurry must be homogeneously mixed to exclude particle segregation, the
liquid must have Newtonian flow behavior, the liquid flow must behave laminar,
and the resulting filter cake structure must be incompressible. If one or more
conditions in this list are violated, the modeling is principally still possible, but of
course more complex. The formation of compressible filter cakes as a practically
very relevant example for the deviation from the ideal boundary conditions will
be examined in Section 6.4.
Figure 6.10 explains the principal situation.
The superficial flow velocity w of the liquid through a filter cake of height hc
and the pressure loss Δp′ can be expressed according to Darcy by Eq. (6.8)
dVL dVL p ⋅ Δp′
w= = = c (6.8)
A A ⋅ dt 𝜂L ⋅ hc
The flow through the filter cake leads to the pressure loss Δp′ in Eq. (6.9)
Δp′ = p1 − p2 (6.9)
The flow through the filter medium leads to the pressure loss Δp′′ in Eq. (6.10)
Δp′′ = p2 − p0 (6.10)
The specific cake permeability pc , which is related to the cake height, provides a
hint, whether the filtration is in a meaningful range between c. 10−11 and 10−16 m2 .
10−11 m2 characterizes very easy to filter slurries, such as coarse crystals, and
10−16 m2 describes extremely hard to filter slurries, such as slimy microorgan-
isms. If the specific permeability is located outside of this range, cake filtration
represents normally not more the adequate separation technique. The equivalent
cake height heq can be used optionally instead of the filter medium resistance Rm .
It gives a better imagination, how relevant the medium resistance is in compari-
son to the cake resistance. The equivalent cake height can be calculated from the
product of absolute resistance of the filter medium and specific permeability of
the filter cake as is expressed in Eq. (6.11)
heq = Rm ⋅ pc (6.11)
96 6 Filter Cake Formation

Normally, heq should result in a few millimeters or less. Equation (6.8) cannot
be integrated in the basic form because the filter cake height is growing during
cake formation and is not constant. Therefore, at first, a correlation between
filtrate volume V L and cake height hc must be found out. This correlation is given
by the concentration parameter 𝜅 in Eq. (2.39). This parameter can be derived
from a mass balance around the total solid/liquid system, which is formulated in
Eq. (6.12)
A ⋅ hc ⋅ (1 − 𝜀) ⋅ 𝜌s = (VL + A ⋅ hc ⋅ 𝜀) ⋅ Y ⋅ 𝜌L (6.12)
On the left side of Eq. (6.12), the solid mass ms of the filter cake can be found and
on the right side the liquid mass mL , which consists of the filtrate volume V L and
the liquid volume, which is enclosed in the voids of the filter cake. The parameter
Y represents, according to Eq. (6.13), the ratio between solid and liquid mass to
make the equation consistent
m
Y = s (6.13)
mL
Finally, the correlation between cake thickness and filtrate volume can be for-
mulated with 𝜅 as summarizing parameter in Eq. (6.14)
Y ⋅ 𝜌L V V
hc = ⋅ L =𝜅⋅ L (6.14)
(1 − 𝜀) ⋅ 𝜌s − Y ⋅ 𝜀 ⋅ 𝜌L A A
Now, the differential equation to describe the growing of the cake can be formu-
lated in Eq. (6.15) and principally be integrated after separation of the variables
𝜂L ⋅ 𝜅 ⋅ VL dVL
Δp′ = ⋅ (6.15)
pc ⋅ A2 dt
However, the filtration takes place not only through the growing filter cake but
also through the porous filter medium with the pressure loss Δp′′ (cf. Figure 6.10).
Also, the flow through the filter medium is formulated in Eq. (6.16) on the basis
of the Darcy equation
𝜂L ⋅ Rm dVL
Δp′′ = ⋅ (6.16)
A dt
To get the total pressure loss Δp of the permeated system, both pressure losses
have to be added, according to Eq. (6.17)
Δp = Δp′ + Δp′′ (6.17)
Now, the final differential equation to describe the filter cake formation can be
formulated in Eq. (6.18)
A ⋅ Δp
dVL = [ ] ⋅ dt (6.18)
𝜅 ⋅ VL
𝜂L ⋅ + Rm
pc ⋅ A
Before integrating Eq. (6.18), the mode of filtration must still be fixed, according
to Figure 6.11.
The filtrate flow is plotted here against the pressure difference. As mentioned in
Section 6.1, cake filtration can be carried out with constant pressure difference Δp
 6.3 Formation of Incompressible Filter Cakes by Pressure Filtration 97

Figure 6.11 Mode of filtration. dVL Constant volume flow

dVL
= f(Δp)
dt Constant
pressure
difference

Δp

or constant volume flow dV L /dt. The process in principle is the same. Although
the pressure difference is held constant, the cake height increases and thus the
flow resistance increases too. As a consequence, the filtrate flow must decrease.
If the filtrate flow is held constant, the cake increases and the flow resistance
increases too. To maintain the constant flow rate, the pressure difference must
increase. If the characteristic of the pump does not provide constant conditions
in a certain range of flow rate or pressure difference, the process follows the pump
capacity curve dV L /dt = f (Δp).
Equation (6.18) should now first be integrated for constant pressure difference.
Equation (6.19) shows the result
[√ ]
A 2 ⋅ 𝜅 ⋅ Δp ⋅ pc ⋅ t1
VL = ⋅ (Rm ⋅ pc ) +
2 − Rm ⋅ pc (6.19)
𝜅 𝜂L
From this result of integration, the typical declining course of filtrate flow with
time can be recognized. If not the filtrate volume but the cake height is interest-
ing, Eq. (6.19) can be transformed with the help of Eq. (6.14), which correlates
filtrate volume and cake height in Eq. (6.20)
√
2 ⋅ 𝜅 ⋅ Δp ⋅ pc ⋅ t1
hc = (Rm ⋅ pc )2 + − Rm ⋅ pc (6.20)
𝜂L
If this equation is solved for the cake forming time t 1 , two terms are formed, as
can be seen in Eq. (6.21)
h2c ⋅ 𝜂L h ⋅R ⋅𝜂
t1 = + c m L (6.21)
2 ⋅ 𝜅 ⋅ pc ⋅ Δp 𝜅 ⋅ Δp
The first term contains the specific cake permeability pc and the squared cake
height hc . The second term contains the filter medium resistance Rm and the lin-
ear cake height. As a consequence, the first term increases much more rapidly
with time than the second one, and if the filter medium is selected well, the sec-
ond term can be neglected more or less from the beginning. In that case, the
filter cake formation can be described according to Eqs. (6.22) and (6.23) by a
very simple equation, which nevertheless is valid for practical use, if the specific
cake permeability is known
√
2 ⋅ 𝜅 ⋅ Δp ⋅ pc ⋅ t1
hc = (6.22)
𝜂L