Low-pressure dewatering kinetics of incompressible filter cakes, II. Constant total pressure loss or high capacity systems

International Journal of Mineral Processing, 5 (1979) 395--405

395

© Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands

LOW-PRESSURE DEWATERING KINETICS OF INCOMPRESSIBLE FILTER CAKES, II. CONSTANT TOTAL PRESSURE LOSS OR HIGH CAPACITY SYSTEMS

R.J. WAKEMAN

Department of Chemical Engineering, University of Exeter, Exeter EX4 4QF (U.K.) (Received July 27, 1978; revised and accepted November 3, 1978)

ABSTRACT Wakeman, R.J., 1979. Low-pressure dewatering kinetics of incompressible filter cakes, II. Constant total pressure loss or high-capacity systems. Int. J. Miner. Process., 5: 395--405. The dynamic dewatering model is extended to systems where the total pressure loss across the cake is constant (that is dewatering is effected by a pump which is able to maintain the vacuum). Using the model, prediction of dewatering characteristics is facilitated when a representative value is assigned to the pore size distribution index. The effects of altering process variables on the decreasing cake saturation and increasing air flow rate are quantified and discussed. The methods presented may be a useful tool for assessing changes in process variables or in the initial sizing of filters.

INTRODUCTION

An important consideration in filter cake dewatering is the mechanism of flow in the voids and the relationships between effective permeabilities and fluid saturations. On a rotary vacuum filter gas, normally air, is drawn through the interstices of the cake along with residual filtrate, resulting in two phase flow conditions. The general problem of two phase flow in filter cakes has been outlined (Wakeman, 1978) and developed for systems involving low-capacity suction equipment. High-capacity systems were considered by BrowneU and Katz (1947) and Lloyd and Dodds (1969) using different bases for their respective relative permeability expressions. The Brownell and Katz (1947) approach is based on the classic conceptions of fluid flow. Just as laminar flow in a pipe can be interpreted in terms of a friction factor which is a simple function of the Reynolds Number, flow through a porous medium can be represented by modified forms of the friction factor and Reynolds Number. The modifications are necessarily related to the porosity of the medium and illustrate the prime importance of porodity to interpretations of flow in porous media. The approach was claimed to be satisfactory for the range of porosity values examined, but some of the

396

porosity exponents become difficult to evaluate outside of these limits. Further investigation and correlation of values was later undertaken by Brownell et al. (1950). Lloyd and Dodds (1969) simply attempted to calculate the flow of liquid from the filter cake by repeated applications of Darcy's law to thin elemental slices of the medium. The method recognised that both permeability and pressure drop are dependent on saturation, the former being obtained from capillary pressure curves in conjunction with a relative permeability model, and the latter being corrected by an effective pressure due to surface tension forces applying at a given saturation. The main simplifying assumptions were a constant displacing air pressure across each slice (with air penetration into the cake having no effect on the total pressure drop) and that the air viscosity is negligible compared with that of the residual filtrate. MODIFICATION TO PROPOSED MODEL

The model used in this paper is a logical extension to that proposed in NOTATION Va ria b le s L P S t U X

o

Cake depth (m) Pressure (N/m 2 ) Cake saturation, fraction of voids filled by fluid Time (s) Volumetric flux density of fluid relative to solid (m3/m:s) Coordinate direction (m) Dimensionless time (defined in Part I) Pore size distribution index

S u p e r s c rip ts *

Dimensionless variable Average value of variable

S u bsc rip ts a

Air

b L R

Breakthrough value Liquid or residual filtrate Reduced value

Part I which involved division of the cake into a number of layers and repeated application of Darcy's law and continuity relationships for both the air and residual filtrate. The model interpreted, with reasonable success, experimental dewatering and air rate curves when data for the variation in total air pressure drop with saturation were used as an input to the computer programme. The equations and conditions used here are identical in every respect except that the total air pressure drop across the bed is now constant, that is, boundary condition (8) of Part I is replaced by: 0~< t ~ tb

x=L

PLI x=L

=constant

tb ( t

x =L

Pa I x~ L

= constant

(1)

397

where the constant is, of course, the initial fluid pressure level at the cake-cloth interface. When expressed in the form of dimensionless variables, the model is now entirely predictive so long as a reasonably representative value of the pore size distribution index X is selected. No data input other than the total pressure drop over the cake and physical properties of the fluids are required. The model should, therefore, be capable of predicting the experimental data of Brownell and Katz (1947). Fig. 1 demonstrates a comparison between the dewatering curves predicted by their semi-empirical model and by the present model, both of which can be compared with the experimental curve. The Brownell and Katz (1947) model apparently gives a closer fit to their experimental data, but in converting the present model from dimensionless to dimensioned variables the Kozeny permeability was used. To use a permeability equation was the only way to estimate the cake permeability in the absence of permeation data pertaining to Brownell and Katz's (1947) experiment, although it has been found preferable to use experimental permeability values (Wakeman, 1978) to convert the variables. In view of this consideration, the fit o f this model to experimental saturation curve was thought acceptable. The model predicted air rates which were a factor 2.5 times greater than the experimental values of Brownell and Katz (1947) over 'the relevant time scale, whereas their model was appartently able to predict rates of 0.8.--0.85 times the experimental. However, such good agreement of the earlier semi-empirical equations with experiment was not obtained in Part I of this study. I

I

0.9 0,8

~c

O.7 o.5

#

oa

&

Q3

.a < cJ

•

~

. . "-.

Present

rectlct/en

Katz

exper;ment

. . . . . . . .

0.1 J

,oo

a

,5o

i

i

:so T~me

(seconds)

too ~,,

Fig. 1. Comparison of Brownell and Katz (1947) data with the presented theory.

398 RESULTS AND DISCUSSION The effects of varying process parameters and of different physical conditions are amenable to analysis using this dynamic dewatering model for high capacity suction systems. The two results of major importance are plotted in each-case, that is, average reduced cake saturation and average d i m e n s i o n less air flow rate to time 0 are both plotted against the dimensionless time 0. Average reduced cake saturation is defined by:

SR = f l SR d(x/L)

(2)

0

and the average air rate to time 0 by:

_1

fo

Y* - - O 0

Va* dO

(3)

The influence of boundary condition (1) is to transfer the constant pressure condition from the liquid to the gas when the gas first forms a continuous phase at the cake-cloth interface. Because of the existence of capillary pressure at this face (as everywhere else in the cake) a minor perturbation is caused in the dewatering curve as the pressure adjusts itself between the two fluids, as shown schematically in Fig. 2. The curve is only affected at short times (generally 0 < 0.06), and the influence of eq. 1 is unnoticable when curves are drawn which cover normal dewatering periods. As the absolute value of the pore size distribution index h is unique to each filter cake (Wakeman, 1976), it is important first to determine its effect on the dewatering characteristics. ~ appears to be independent o f mean particle sizes, but is probably affected by packing arrangements within the cake. A representative value, based on numerous experiments, is about 5. Figs. 3a 1

l

o.~ 0

0.7

Fig. 2. The effect of boundary condition (1) at short times.

399

7

p*

= fO0.O

aO, t p~ = 9GO aL,l

i O.B

7~=9

¢k3

o

0

i

, ,]5

,

, I

,

i

,

i

L

1.5 Oimens

Fig.

i

i

2.5

,

,

,

,

onLess

Tlme

i

.

.

.

.

J

1

=45

3

--

3. The e f f e c t o f X o n saturation (a) and air f l o w rate (b).

and 3b show that whereas dewatering curves are unaffected by ~, slightly greater air rates are obtained with higher X values. The value o f 5 was assumed in subsequent computations as this appeared to be a reasonably representative value on the basis of many experiments and gave a fair interpretation of air rate curves in most instances during Part I of this study. The principal dimensionless parameters which can be varied and also have some bearing on filter operation are the vacuum level and the viscosity of the residual filtrate. It is recognised that many other parameters can be altered under process conditions, but these appear in the dimensionless parameters selected for the computations and the effect of varying any of these can be interpreted from the general curves presented here, once further information about the particular filtration is known. The total pressure drop across the bed governs the rate at which liquid is withdrawn, but both the pressure drop and the true pressure level control the air flow rate with higher average rates being obtained at lower pressures. The effect of the total pressure drop on cake saturation is shown in Fig. 4a, and on air flow rate in Fig. 4b. These curves simply indicate the expected changes in saturation and air rate as the driving force is increased. At a higher dewatering driving force, the initial desaturation rate is greater and the saturation levels o f f more rapidly, zero average reduced saturation (that is, the irreducible saturation) being approached asymptotically. Fig. 5 shows the marked effect o f filtrate viscosity on the reduced satura-

400

tion curves. Since the filtrate viscosity is included in the dimensionless time 0, its effect is not clear unless real time is plotted. To do this some realistic parameter values were chosen, and it is apparent that more viscous filtrate is displaced from the cake at slower rates as would be expected. The cake saturation is quite sensitive to small variations in viscosity, especially during the initial stages of dewatering. It is, of course, c o m m o n industrial practice to reduce the liquid viscosity prior to filtration, b u t effects of this have not hitherto been amenable to prediction in any quantitative way other than from the results of many experiments. Apart from the decrease of average cake saturation during dewatering, the development of saturation profiles throughout the depth of the cake is of fundamental interest. Limited experimental data are available (Wakeman, 1976) to indicate that liquid is distributed throughout the depth o f the cake at its irreducible saturation level, the moisture content being the lowest at the air inlet face. In fairly high-permeability cakes the residual moisture content was found to be almost independent of cake depth, but in high specific resistance cakes the saturation level tends to increase to a maximum at the fluid outflow face. Such data, however, were necessarily obtained after a finite dewatering period. Fig. 4a shows h o w the approach to the irreducible C.8

0.7 "i

-'~

y~/~ ~ o.o~8~,6

0.5. I

A

!1' c ?

= 5

P2 : ~oo.o

0.5

E

\

m

~ 0.2

~

&5

I

× ~ ~ - - ~ - x

1.5

2

2.5 Dimensionless

3 Time

Fig. 4a. The effect o f dewatering pressure on drainage curves.

~5

&

~.5

401 7L

/

12

Legend as for ;-/g. 4a

I0 / 0

i

_ ~ x J -

-

J J

o

/

eo E

j

~

//

&

II-

×

2

J

i

i

C,5

i

1

i

1,.5

i

I

I

2

i

i

2.5

i

i

3

Dimensionless

1

3,5

Time

i

i

i

&

i

~5

T - -

Fig. 4b. The e f f e c t o f dewatering pressure on air f l o w rate curves.

=

k *~

.

,

~

GL

.

o,t

d = 5o F L : U.U5 r#

@.3

g

0.2

o

0.1

(1001

J

,

30

J

,

60

,

i

90

,

_

12(J

,

,

,

150 f/{

180 -St~}

Fig. 5. Effect o f liquid viscosity on dewatering curves.

,

,

210 (seconds}

,

2&O

,

°

270

,

300

402

saturation level is asymptotic, when the minmum moisture content will be distributed uniformly only at an infinite time. In the case of incompressible cakes with a uniform porosity distribution, it is reasonable to presume that the residual liquid distribution eventually reached will also be uniform.

....

/

~/~L.ozs~6

o.9 "qt

(

/

h/

~.

y

°.8

.

~.+~,,~//

,'-....

_ o.o33 " ~

0.0711 O.L

o.1

~

~

4.9029 - ,

0

.-----------

14.8322 -

I

I

I

I

I

i

I

I

I

0.1

0.2

0.3

0.4

0.5

0.6

0.?

0.8

o.g

Free surf(Ice

C(lke/ctoth Dimensiontess

Distance in

interface

Cake {x/L)

Fig. 6. The d e v e l o p m e n t o f saturation profiles within a filter cake.

403

0.9

p~,, = 100.0 O,t p* = 80.0 a L,t

0,8

02

g

0.5

22f

~d

~35 I 0,5 D

,0%2 /

0.4

0.3

,,

x

~

uJ769

o.3868 0,2

~_.___---

0.7016

0.1

0

t

¢

I

!

!

03

0.2

0.3

0.4

....

I-.

05

I

0.6

!

0.7

Free surface Dimensionless Distance in Cake (~/L)

i

0.8

i

0.9

t

Cake/cloth interface

Fig. 7. The d e v e l o p m e n t o f saturation profiles within a filter cake.

Fig. 6 shows the saturation profiles through the depth of the cake as they develop, and Fig. 7 shows the results o f similar calculations for a more viscous liquid. At zero time the pressure loss is entirely through the liquid filling the cake voids, but at short times a fairly large proportion o f the

404

liquid is displaced from the voids close to the air inlet surface to leave air as a continuous fluid in the evacuated volume. A small fraction of the dewatering pressure is loss through this air, and there is a correspondingly small pressure loss through the retained filtrate up to the air " f r o n t " . The vacuum (pressure) level is constant, however, and the resulting effect is an increasing pressure gradient through the saturated part of the cake, which causes this liquid to accelerate until air breaks through to the filter cloth. After air breakthrough, the pressure loss is entirely due to flow of air through the voids, and the liquid pressure distribution is governed by the capillary pressure profile (which is in turn a function of the cake saturation profile). Shortly after breakthrough, the voids close to the fluid outflow face are relatively highly saturated (compared with those close to the air inlet face), giving a high pressure loss in both the air and liquid phases. Although these voids are being supplied with draining liquid continuously, they dewater more rapidly to make the saturation go through a maximum close to the air inlet face. After a while, when the voids have a fairly high air saturation throughout, the dewatering rate becomes more uniform throughout the cake depth and decreases according to the reduction in the effective liquid permeability which is itself distributed through the voids. Comparison of Figs. 6 and 7 shows that a steeper saturation gradient exists close to the air inlet surface when the liquid is more viscous, causing the saturation profiles to be more pronounced. Also, that the bed is more saturated at any given time 0 when the liquid is more viscous, as would be expected. In both cases the reduced saturation rises very sharply from zero at the air inlet face. The reason for this is that there is no liquid flow into the cake, but this can only happen when effective liquid permeability is zero, which is only true when the reduced saturation is zero. To avoid unnecessary confusion of detail, the profiles close to the air inlet surface have been omitted from Figs. 6 and 7. CONCLUDING REMARKS

The dynamic model for filter cake dewatering has been extended to situations when the vacuum pump is operated such that it has adequate capacity to maintain a constant pressure drop across the cake. The dewatering curves are not sensitive to the pore size distribution index, and when a representative value is assigned to the index, the model is capable of predicting dewatering characteristics. The effects of driving force and filtrate viscosity upon the moisture content of incompressible filter cakes have been demonstrated. The effects of other variables can be assessed by further analysis of the curves presented. The model may prove useful in the evaluation of changes in process variables or in the initial sizing o f filters, when dewatering is being accomplished by either a constant pressure or a constant vacuum.

405 ACKNOWLEDGEMENT

The author wishes to record his gratitude for the receipt of a Science Research Council Grant to support this work. REFERENCES Brownell, L.E. and Katz, D.L., 1947. Flow of fluids through porous media. Chem. Eng. Prog., 43: 703--712. Brownell, L.E., Dombrowski, H.S. and Dickey, C.A., 1950. Chem. Eng. Prog., 46: 415. Lloyd, P.J. and Dodds, J.A., 1969. The dewatering characteristics of packed beds and filter cakes. Presented at 64th National Meeting of A.I.Ch.E., New Orleans. Wakeman, R.J., 1:)76. Vacuum dewatering and residual saturation of incompressible filter cakes. Int. J. Miner. Process., 3: 193--206. Wakeman, R.J., 1979. Low-pressure dewatering kinetics of incompressible filter cakes, I. Variable total pressure loss or low-capacity systems. Int. J. Miner. Process., 5: 379--393.(this issue).