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44 Filtration

Fundarnenfals

2.5 Forms of Cake Filtration Equation

Substituting Equation (2.16) into Equation (2.12) gives:

dV A AP

dt pcVa

(2.19)

Equation (2.19) contains three variables and four constants: time, filtrate volume and

pressure; and filtration area, viscosity, concentration and specific resistance. The last two

are constant only

if

the filter cake is incompressible. The equation can be solved

analytically only

if

one of the three variables is held constant.

This

reflects the physical

mode of operation of industrial filters; vacuum filtration tends to be at constant pressure

and pressure filtration

is

oRen under constant rate, at least until some predetermined

pressure has been achieved. Thus the following mathematical models are very relevant to

these filtrations.

2 5 1 Constant Pressure Filtration

Under these conditions Equation (2.19) can be rearranged and integrated as follows:

the limits are given

by:

zero filtrate volume at zero time, V volume filtrate after time t,

thus:

(2.20)

In deriving Equation (2.20) any pressure loss due to the flow of filtrate through the

6lter medium has been neglected. This assumption can be removed by assuming that the

pressure drop in the medium

APm

can be added to the pressure drop over the filter cake

h p o give the total or overall pressure drop:

A P = APc APm

(2.21)

Darcy's law can then be applied to both terms:

(2.22)


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2.5

Forms

of Cake

Filtration Equation 45

where Lm nd

k

are the medium ,depth and permeability, respectively. If the medium

resistance and depth remains constant during filtration these

two

constants can be

replaced by a single constant known as the medium resistance , with units of m- :

This expression can be substituted into Equation (2.22), which can then be rearranged

and integrated:

After integration and rearrangement the following equation, known as the linearised

parabolic rate law results:

@m

v -

V 2A2AP

A A P

(2.23)

Equation

(2.23)

is a straight line, where

t/V

is the dependent and V is the independent

variable. Thus a graph of the experimental data points of

t/Y

against V permits

calculation of the gradient and intercept of Equation

(2.23)

muth,

19351:

The gradient and intercept are as follows:

lca

Gradient = ___

2AZAP

and

P R m

Intercept = _ _ _

A A P

(2.24)

(2.25)

Thus if

the liquid viscosity, filter area, fitration pressure and

mass

of

dry

cake per unit

volume of filtrate, either fiom Equation

(2.17)

or

(2.18),

are known, the graphical values

can be used to calculate the cake specific resistance and filter medium resistance.

Worked

Example: the data shown

in

Figure 2.7 were obtained from the constant

pressure period of a pilot scale plate and .frame filter press. Calculate the cake

resistance given: filter area 2.72 m viscosity 10 Pa s,

mass

of dry cake per unit

volume filtrate 125

kg

m-3 and filtrate pressure 3 bar. The specific resistance by

Equation

(2.24)

is

5 .4~10

m kg-. The apparent medium resistance is

2 .9~10

m-

by Equation (2.25). However, in this instance the medium resistance is a composite

term including the resistance to filtrate

flow

due to the cake formed during the


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6 iltration

Fundamentals

preceding constant rate filtration period on the filter press, in addition to the true

medium resistance.

3E

6000

5000

0

+

? 4000

0

.r 3

2. 2000

c

c.

a

0 024

0 039

327 0.071

418 0.088

472 0.096

538 0 106

0

0.02

0.04 0.06

0.08 0.10

0.12 0.14

I

.+

Volume Filtrate Collected, m3

Wgure 2.7 Linearised parabolic rate law plot

t

is important to realise that the above equations are applicable to both small-scale

laboratory test data and to fdl-scale industrial filters operating under conditions of

constant pressure filtration. Thus specific resistance and medium resistance are regarded

as two design variables which can be used to optimise filter throughput, or to scale up

laboratory data, as

will

be discussed fhther in Chapter 11. Further important details

regarding the filter medium resistance are discussed in Section 2.9, and these should be

read before any further conclusions are made regarding thisvariable.

There are various forms of the parabolic rate law in common use. One alternative is to

plot tHl)/ V-VI against Vl where tl and Vl are some arbitrary datum values of time and

filtrate volume. This is usefid

if

a long time has been taken in reaching the h a 1 and

constant value of pressure across the cake and cloth.

It

is

also

usefd

in the analysis

of

data in which a step change of the filtration pressure was performed, such as detailed

in

Section 2.6. The value of the medium resistance calculated fkom Equation (2.25) under

these circumstances, has dubious meaning, however, as it represents both the medium

resistance and the resistance due to the cake deposited prior to the datum. Some

investigators work with the differential form of Equation (2.23), plotting the reciprocal

of the instantaneous filtrate rate q against volume of filtrate, which again produces

a

straight line in accordance with equations (2.26) and (2.27):

(2.26)


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2 5 Forms

of

Cake Filtration Equation

47

i.e.:

1

v -

9 AA?

go

_ = -

(2 .27)

where

4

is the filtrate rate at the start of the filtration when no cake is present. It is not,

however, the same as the medium resistance

n

the total absence

of

slurry to be atered, as

will

be discussed in Section

2.9.

Use of Equations

(2 .26)

and

(2 .27)

presents some

practical difEculties as the instantaneous filtrate flow rate usually has to be determined by

graphical merentiation (i.e taking tangents) of the filtrate volume-time curve. Such a

procedure is notoriously inaccurate.

Finally, one method which is readily applied to computer spreadsheet use is to consider

the incremental version of Equation (2 .23) . The sltration data is arranged into equal

volumes

of

filtrate, and the corresponding time for each increment is calculated. The

filtration

starts

at

to

and

YO,

the next increment

is

tl

and

Vl ,

and subsequent times are

measured after equal volumes of filtrate ( A v ) have been collected. The incremental

equation for the first increment can be represented as:

t , =

a(AV)

+ b ( A V )

subsequent increments occur at (2AY), (3AV), etc.

If the difference between consecutive increments is taken then the resulting, and general,

st

Werence equation is:

At ,

=

( 2 1 2 - l ) a ( A V ) + b ( A Y )

The second-Werence equation is obtained om the difference between two

consecutive fist Werence equations, this

will

be:

2a(AV)

Thus the second difference in the times required to achieve the filtrate volumes can be

equated with the above, and rearranged

to

provide values of a and ultimately specific

resistance.

This

technique is best illustrated by the application of a Mereme table, which

is given below.

The equation or

b,

and hence R s more complex, but again does not require any

graphical construction.

The advantage

of

the above procedure is that values

of

both specific resistance and

medium resistance can be calculated for each data point (or at least n-2 data points where

is

the total number

of

points). This helps to highlight any erroneous data points, and

provides some indication

of

the spread of experimental values for specific and


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48 2 Filtration Fundamentals

medium resistances. Clearly, the mean value of specific and medium resistance can be

used in further computations, possibly combined with the lowest and highest values

taken fi om he difference table as an indication

of

this measure of spread on these values.

The routine appears very complex on first viewing but lends itself to computer

spreadsheet application with, therefore, the minimum amount of repetitive calculation

once set up.

Table 2.1 Difference table for constant pressure filtration

Incremental equation First difference Second difference

t l =

a(Av2+b(AV)

tz= a(2AQ2+b(2AV) At2= 3a(Aq2+bAV

t3

=

a(3AV2+b(3AV) At3= 5a(AV)+bAV A(At)

=

2a(AV)2

. . .

....

....

tn-l

= (n-l)2a(AV)2+(n-1)b(AV) At,,-l

=

(2n-3)a(AV)+b(AV A(At)

=

2a(AV)

t,,

=

na(A V)+nb(A V

=

(2n-l)a(AV)2+b(AV) A(At) = 2a(Av2

2.5.2

Constant Rate Filtration

This type of filtration commonly occurs when an efficient positive displacement pump

is used to feed a pressure filter. The pump delivers a

unifoim

volume of slurry into the

filter; hence the filtration rate remajns constant when filtering an incompressible material.

In

order to achieve

this

constant rate the pressure delivered by the pump must rise, to

overcome the increasing resistance to filtration caused by cake deposition. It is usual for

such a pumping system to include a pressure relief valve;

if

the filtration cycle is long

enough, the constant rate period

wiU

be followed by

a

constant pressure period after the

pressure relief valve opens.

Constant rate filtration is easily observed

on a

plot of filtrate volume against time, as

illustratedin Figure 2.8.

Under these circumstances:

= Constant

d V Y

dt

t


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