Selecting the right centrifugal pump for a pressure filter

By F NI Tiller Chemical

Engineering

Dept, University

of Houston, Houston,

TX 772204-4792.

USA

and C S Yeh Mobil Research

& Development

Corp, 13777 Midway,

A method is presented for selecting a centrifugal pump to produce a specified cake thickness within a desired time frame while operating close to the best efficiency point (BEP). The procedure can be used for either a fixed or variable throV!le between the pump and the filter. A parallelogram whose sides correspond to ranges of thickness and time is drawn on a logarithmic plot of the pump characteristics. If the BEP falls within the thickness-time parallelogram, the pump is considered to be satisfactory. SeLacrloN OF a pump lo fit a SC1 Of specifications for filtralion in a filter press or a pressure vessel (leaf and tubular) filter is an important step in design or retrofit. For an existing pump and filter combination., estimation of changes in time. thickness. and prcssurc for slurrlcs with varying properties is frcqucntly required. Similar tcchniqucs can bc employed for both selection of pumps and determination of the effects of new slurries on process conditions. Pumps employed in filtration can be classified as (I) centrifugal. (2) progresslvc cavity or screw pump (Moyno or Mono), (3) diaphragm. (4) positive displacement. and (5) comprcsscd gas cithcr alone or combined with a pump. Each typo posscsscs its advantages and disadvantages. Cetrifugal pumps are widely cmploycd in spite of rhcir tendency to dcgradc aggregates because of large shear forces rcsulring from high velocity ~mpcllcrs. They suffer in that flow rate dccreascs and pressure rises as cnkc builds up. Characteristics of ‘.ypic:i pumps arc sh-wn in Figs 1A and 8. The cncioscd arca with each pump represents a range of characteristics available with different sized casings and impellers. Similar curves also exist for different speeds with a given impcllcr size. Fig IB shows the characteristics of a given pump (3in inlet and 2in outlet) with diffcrcnt sized impctlcrs. Lines of constant efficiency arc superimposed on the characteristic curves. All of the pump curves are truncated at a cut-off prcsqurc equal roughly to OS-O.75 of the maximum prcssurc. A pump should not be operated at high rates below the cut-off pressure or at low rntcs cncountcrcd at the other extremes of the characteristic curve. close to the best Pumps arc desi ncd 10 give smooth operation efficiency point ?BEP). If a pump is operated at high flow rates and low heads. surges and vibration with low cncrgy cf’ficicncy can lead to problems. When il pump operates on the high end of the curve. the an@ at which the liquid exits the impcllcr is not in accord with the dcslgned vane angles resulting in cddles which cause erosion on the vanes and impellers. At low flow rates. dctcntion time is markedly increased leading to heating and increased possibility of aggrcgatc degradation. It has been previously suggested (‘) that the output of a ccnlrifugal pump could be regulated by throttting the discharge. Although that can be done without danger of immcdiatc damage 10 the pump. manufacturers do nor rccommcnd operation at less than 20% of maximum capacity. A bypass valve can be usefully employed to maintain totai Ilow close to the BEP thereby avoiding operation at too low a ralo. A system with a throttle A and bypass B is illustrated in Fig 2. In the simplest tvpc of operation. A is throttled as suggcstcd”’ so that operation starts at point A in Fig 18. As the cake resistance builds up, opcralion proceeds with pressure and rutc following the characteristic curve. In another mode of operation. A is throttled to the BEP at zero time. Then as cake resistan:: builds up. B is opened so as to maintain pressure at the BEP. The rate remains constant. and pressure drop across the cake builds slowly. Once the throttle is open complctcly. the pressure and rate follow the characteristic curve. The pressure could be maintained at the BEP by opening the bypass B once A was wide open. Such a procedure amounts to a constant rate followed by a constant pressure filtration!“. Fig I is a small sample of the large number of pumps available for selection. The first step in looking for a pump with the appropriate characteristics involves establishing ranges of desired thickness and time. If a fitter is already available, there may be quite a narrow range of satisfactory thickncsscs. Plate-and-frame presses. rcccsscd plate filters. availability of mcmhranc squeezing. vertical and horizontal leaf fitters. and candle filters all present varying rcquircments for cake thickness. Cake formation time for maxitium c~clc output should be considcrcd in establishinS the desired cycle’. ‘. As a crude gcncrality. the form time should be Somewhat larger than the dead time in which no cake is being produced. Fig 3 prO*JldeS an example of calculations involving scvcral pumps with respect to a desired range of thickncsscs and cake formation times. In gcncral, difficult IO filrcr materials rcquirc thin cakes which may bc under 1Omm while easy IO filter 48

Dallas,

TX 75287,

USA

slurries may result in cakes as thick as 100mm. Before dcvcloping techniques for selecting centrifugal pumps, WC first discuss methods for predicting volume and prcssurc vs time for constant rate filtration and variable pressure-variable rate operation.

Material Balances Mathematical manipulation of equations in the theory of filtration rcquircs formulas relating volume v 0r filtrate/unit area. volume w, 0r dry solids deposited in the cake per uni’t area. and cake thickness L which also represents the total cake volume/unit area. Assuming Gh = volume fraction of solids in the slurry and E,;,, = average cake 240

80

GALLONS

PER

MINUTE

Fig 1A. Ranges of different sized pumps”’

P

IMPELLER

200

0

SIZE 2 x3-6 EFFIECIENCY

100 GALLONS

Fig IB. Pump characteristics

st--a:-.-.....Y

200

300

PER

MfNUTE

0 “es: Efficiency Point”’

I

Fig 2. Location of throttle and bypass valves JanuarylFepruary

1990 Filtration & Separation

APt-h --$-

L=(&)p 3x4 MT 1750 RPM

r

-

t

2x3 6MT 1750 RPM

If the thickness L is specified. then AP, and q as taken from the pumpcurve must satisfy Eqs (9) and (IO). In constant rate filtration, v = qt. * and the time can bc c:~lcula~cd by: t=v/q=(F-l)L/q (11) Combining

Eqs (9), (IO) and (11) products: t = (F - 1) (K,,,.Iu) (AP,Jq’)

(12)

AP’+ q’

(131

t=(F26

40

60

80

100

120

MINUTES Fig 3. Limits on time and thickness solidosity (vohtmc fraction of solids in the cake). c uating the volume tltrate lcads to: of slurry to the sum of the volumes of the cake and ‘E. w, -- _

s+v=L+” a,;,,*

4%

The quantity v has dimension ing these equations produces: F%V

w, =

(1)

of volume/

area = Icngth.

Rcarrang-

= cv = E.,,,L

(2)

where F = E,&+,. The average cake solidosity is a function of the pressure drop AP, across the cake. However, to simplify calculations. F,;,,,. F. and C will bc assumed constant and equal to their values at the final AP,. Rate Equations The instantaneous flow rate per unit area q is given by: + R,)

where q = dvldt. Eliminating relations for v and L yields:

w, from

Eq (3) and

establishing

t=

Combining

dvlq

5 ($. + J-)

(14) by: (15)

=v/q,,M

mean rate qtthl’J). Eq (1.5)

Eqs (9). (15) and (16) leads to: t=(F-

l)(K,,,.Iu)

o,,* = a(1 - n)APE

(7)

K;,,. = I%

(8)

AP,” with Eqs (3)-(5).

Constant Rate Filtration Combination of Eqs (4) and (5) with Eq (6) gives: L _ K;Y “9”

(9)

at the BEP. WC note that appear in this formula.

(17)

s q2q11hl

t = (F - 1) (K,,,.Ip) ‘s

t18)

AP,

If K,,, from Eq (8) is used. Eq (17) becomes:

1

(6)

E

where AP, and q correspond to conditions the two groups (K&L) and (AP,Jq) Substitution of Eq (8) for K,,, leads to:

V t-

(4)

Apt’

will be used in combination

Centrifugal Pump Filtrafiqdy) WC shall consider t c cast u-t whtch a throttle placed hctwccn the pump and filter is adjusted so that the initial flow rate corresponds to the cut-off pressure point T in Fig IB. We shall assume that the pressure follows the characteristic curve to some arbitrary point above the BEP. For example. in Fig 18, the filtration might start and finish at the (10% efficiency points. Point T would then bc moved up from the cut-off point. Operation would be carried out in the (,I)-70% cfficicncy range. Eq (9) scrvcs for any point on the characteristic curve as wull iis the BEP. If the terminal oint is lahellcd ‘2’. then AP,.2 and q2 would bc substituted into Eqs ( 8 ) and (10). When the rate varies with time. t is calculntcd by means of:

This is equivalent to using an harmonic can also bc written as:

Constitutive Equations Local values of E.. LX: and K required for the theoretical developments are provided m the appendix. For illustrative purposes, a simplified set of average values for moderately compressible matcrials wiil be used with the following formulas:

These equations

Two new grou s (F - I) and (APdq’) ttppcitr in Eq (12). With four groups. two of w R-tch depend on pump characteristics, it is possible to pick a pump which will satisfy the L-t rcquiremcnts. The equations in this section apply to a ccntrifugill pump that is throttled with a variable resistance which leads to a constant rate of filtration. They can also be used to specify the rate-prcssurc relationship for a constant rate pump.

(3)

dL q = (F - ‘) dt = p(LIK,,: f R,)

= B1-”

1) (,_h)t,

A plot of I/q vs v is nearly linear, and t CiIn bc approximated

dv P q = ;II = u(cy,,,Cv + R,)

Lv

J

s0

P = p(LfK;,v + R,,,)

P q = u(o,,,w,

(10)

*n-%,--h 19)

Normally there would be little difference in comparison to Eqs (18) and (19).

in using Eqs (12) and

13)

Example With Variable Throttle A variable throttle will be introduced bctwcen the filter and pump so that constant rate operation at BEP can be carried out. An aqueous slurry at 23°C with IL = 0.944mPa - s. p,_= 997kglm’. and containing 2.0% by volume of solids (sp.gr. = 1.36) is to be filtered in a 50m’ horizontal leaf filter in which the range of thickness to bc investigated is 50-100mm of cake. The formation time should lie Medium resistance is estimated IO be between 30-120 min. l.t%(EIO)m-‘. The average permeability and solidosity are given by: K;,, = 3.09 (E - 12) AP:“.‘s The units are m’. E,;,, = O.l76AP’!,t” ‘

Table 1. Pump Characteristics Pump

2X2‘ 2x3 3x3 4x4

pt.2 971 184 926 254

PiW$ % 651 215

where AP, is given in kPa. The average specific resistance

PI.1

43 q-10mIm PI.2 BEP

836 100 :::

$:! “2:::

::3: 5.61 4.04

P .s

pt.1 Z:% 6.44 5.87

refer to tge inlet and outletdiameters in inches (Goulds Pumps, 1977). The values of q in m3/m . s were obtained by dividing the pump rate in m /hr by 50 - 3600 where 50 = filterarea in square meters

‘Numbers

Filtration & Separation

January/February

1990

%V = l/K,,,r,,,,,

is:

= 1.84(E12) AP;,‘”

The units are m-’ E s (20)-(22) Below IOkPa. K:,, = d.7lqE - 13)m-. The solution requires that a,i,v at tS?,ottle rcsistancc has been reduced across the cake. Assuming that

are used when

AP, L IOkPa.

the BEP be known when the to zero and the full pressure is the pressure will be in the 49

ncighhourhood of’ ZOOkPu. Eq (21) yields E,,, = 0.350. Then F = ’ 0.350/0.0Z = 17.5. Four umps will hc invcstigatcd. The conditions at points I. 1. :tnd the B B P for the pumps urc listed in Tublc I. For simplicity. point I has been chosen to correspond to T. Solvine for AP,!-’ in Eq (IO) und substituting the uppropri:ttc values To? the puritmctcrs yields: Ap,,

L

h

_ _

(1 -Lc~- 3) It

5

J

Ap:.Js

a

0.45 .0.944

1500 1000 I

. (E - 3)

l.OOO* l.N*(E

- 12) Lq ‘23’

600,

whcrc I .(I00 is the conversion T;lctor from P:I to kPu. Suhstitutine 0.05 and 0. It) for L yields IWO limitin_c cquutions involving AP, mid qc Api.’ Rcpcuting

= I .57X (EJ)q or 3.056 (E-l)q

the proccdurc

Substituting

400

(24)

for Eq ( 13) Ic;~ds to:

t = I A00 und 72(lOs products: AP$.‘,’ = 3.33J(E7)q’or

l3.34( E7)q’

100 80

(26)

Eqs (24) and (Zh) provide limits on the Iocation of the BEP. For purposes of :m:llysis. Eqs (33) und (25) ;lrc plilccd in lo?l,arithmic form its follows amI iirc plotted on top of fhc pump chunlctcristics in Fig -I: log AP, = 2.22log q + 7.22lop

The douhlr

v;~lucs

correspond

l.S%(E-i) 3.056( E-l)

60 40 30

(27)

min.

20

to S.0 unrl IOcm thickncsscs. 3.3S(E4) 13,34(E7)

log Al’, = -1.4-llog q -I- 2.Ztog

(28)

10 8

The two vulucs corrcspcmd to times of 0.5 and 3.Ohrs. At the start of filtration. the throftlc plus the medium rcsistnncc cmlblcs operation to hc cffcctcd at the BEP (or ;Iny other point on the characteristic curve). As the throttle rcsistancc is rcduccd. prcssurc drop across the c;~kc incrcuscs. When the throttle is wide open. the rcmuinin$ rcsistancc is due to the medium und piping. WC ncglcct those rcslstmiccs und wx the prcssurc at the BEP as equal to AP,. Plots of P vs q ut constant L and t in :Iccord with Eqs (27) and (78) arc supcrimposcd on the chzlrnctcristics of the p~mips uppcaring in Tuhlc I. The lines rcprcscnting constilnt times of 30 min. 1.O and 3.Ohrs have twice the slope of the two constant thickness lines. To satisfy the ori_cinul spccific;ltions. the BEP should lit inside the diamond sh:lpcd. sl~dcd rcpion. If the cxuct cquutions in the appendix wcrc cmploycd. the Iowcr portion of the diamond shaped region wvoulcl hc modified. Any ch;mgcs rcsultinc? from the use of more :lccuratc cquutions would bc of no practi&l importance. The BEP’s ror the 7x3 and 4X-I pumps szrtisfy the rcquircmcnts. Choosing the 1x3 pump. the following calculations can hc made:

6 4 3 1’o-4

(E - -1) = 0.0671~~

t = S.40 (E - 5) . 145”~.‘%i.h(E - 4)’ = 1.3’)6scc Corresponding Other conditions pump sclcction.

34

Flow

6

8

dO--3

m3/m'-s

Rate,

P 100 Fig

L = 3.17 (E - 6) * 145”?4.6

2

a

l

200

..m,,

400

4. Constant Land t lines superimposed

800 on pump characteristics

(29) (30)

vulucs for the 4X4 pump ;Lrc 0.0907m und 3.7OSscc. within tho pcrmissiblc region ;Irc of intorcst in

Range of Suitable Pumps Any pump whose chnractcristics fall within the shaded ;Irc:l in F.ig -I coutd hc considered. Even when the BEP might not lit in the dcslrcd rcpion. other portions of the curve could hc cmploycd with the nppropriatc throttling rcsistancc. Three points ma&cd A-C at widely different locntions arc shown to lx within the pcrmissihlc pnr:rllclogram. The conditions ‘0’ cncl3 of those points ;Irc summoriscd in Table 2. The range of suit;tblc pumps which lit in the pur;~llclogr;un is quite

POilll

kPa

A

500

:

:x

Table 2. Suitable

Pumps

HeadJt

GPM

qmls 6 0 (E - 4) $$I:;

L.cm

460

I:4 67

l.min :A

:%

a7

200 RANGE OF FLLTRATION

PUMP

-

CHARACTERISTICS 2-3

larec.

athcr stud& can bc made with the information av;lilablc in Fig 4. For cxamplc. il’ a prcssurc of 30OkP:1 wcrc to bc used. Eqs (23) ;Ind (2.5) would give: qL = J.ZhXE (- 5) (31) q’t = 7.038E (- 4)

(32)

From Fig 4. it can bc seen that the thinnest cake will hc grcatcr than 5cm and the longest time will bc about one hour. The r:ltc varies from 4.3-6.3 (E - 4)m/s (3SO-SOOgpm). The corresponding thickness and time variations arc A.3-K.9cm and 4l-9Xmin. Example with Fixed Throttle When ;I f’ixcd throttle iscmploycd. the prcsssurc drop availnhlc to the cake dcpcnds upon the flow rate. Assuming frictional losses through the throttle are fubulcnt und losses through the medium urc viscous leads to: (33) AP,. = P - VqR,, - (P, - LtqtR,,,) (q/q,)’ 50

ApfhlEDIUM)

Oh 0

.OOOl

Fig 5. Pressure

.0002

.0003

.0004

.0005

.o

6

FLOW RATE, q. m3/m2.s drops across cake, throttle, and medium

January/February

1990

Filtration & Separation

To illustrate the method, the revious cxamplc will hc rcworkcd ? with I fixed throttle between the _%3 pump nnd the filter. WC nssumc that the maximum cake thickness is 7Smm in the SOm’ filter nnd dctcrminc whether or not the rcquircd time is rcnsonnhfe. Medium resistan’cc cquuls 1A8 (El(l)m‘. Assuming that poinr 1 in Eq (33) lies at the cut-off point of P = IOOkPo and q = ~5 (E - d)mls. the prcssurc drop ucross the cukc bccomcs: Table 3. I. and v vs q in Filtration With Fixed Throttle P.&Pa

AP..kPn

t-m

v,m

100 114 140 126

*E3 80.3 565

:027 0051 0040

E.344 0 557

142 155

108 a4 0

a 053 067

: 3zi: 022

166 1;:

129 149 164

0 083 C%

2.090

I 1291 1 642

hP,=P-O.Wl(E--3). zp-

January/February

1990

vs v

(34)

from Eq (2) us follows:

On the basis of volucs of q and AP, obtained from Fig 5. the volumclunit urcu v and L can IX c;tlcul;lted. Summnry in Tuhlc 3.

-

40

.20 Time.

Filtrate Volume \I, m3/m2

Filtration & Separation

2.51 (EK)q2

The volume of filtrntc cun also hc obtuincd

0

of time from plot of l/q

-

Omission of the medium resistance would have un inconscqucn~ini cffcct on the results. The pump chaructcristics nnd the prcssurc drops across c;lkc. throttle. and medium nrc shown in Fi * 5. At cnch oint on the chnrnctcristic curve. the tt *lckncss is fixctl by Eqs (9) un s (IO) or their cquivalcnt. Eq (23). Solving for I, in Eq (73) yields: L = 3.27 (E - 6) AP,.“+q (35J

0.10

Fig 6. Determination

I.6K(ElO)q/l.Ol~o--.OS(q/h(E-d))’

i.W(W)q

60

minutes

Fig 7. Thickness, pump pressure, and pressure drop across the cake as a function of time

51

:: B a S e d upon t h e ' v a l u e s in Table 3, the final p u m p pressUre Would bc about 160kP. A t that point, the cake thickness would be 75ram: In o r d e r to obtain the time, a plot o f I/q vs v is c o n s t r u c t e d in Fig 6 in accord w i t h Eq (14). Results o f the numerical calculations =ire s h o w n in Fig 7 w h e r e thc cake thickness L . p u m p p r e s s u r e P, and p r e s s u r e d r o p across the cakc are shown as functions o f time. : " T h e results o f t h e e x t e n d e d c a l c u l a t i o n s b a s e d o n T a b l e 3 can be a p p r o x i m a t e d b y m e a n s o f E q s ( I l l ) a n d 119). F o r a c a k e w i t h L = 75mm; Eq ([0) becomes: z~t~#,!'4'~/q = 2,292 ( E 4 )

(37)

C o m b i n i n g this eCluation with Eq (34) and the p u m p characteristics yields an answer of P = 160.SkPa, q = 3.75 (E " 4)mrs, and Ap~. = 12i).3kPa, T h e h a r m o n i c m e a n rate is t h e n given by: grim

= 2 • 3 . 7 5 • 6 . 0 ( E -- 8 ) 1 1 3 , 7 5 + 6.0) ( E - 4 ) = 4 . 6 I ( E -- 4)m/s

(38)

t = 5.40 ( E - 5 ) A p ~=rT . '~/qlqHM = 5 , 4 0 ( E - 5) • 120.3u'4-~/3,75 • 4.61 ( E - S) = 2 , 6 9 6 "

(39)

T h e time equals 44.9rain as c o m p a r e d to the m o r e exact answer o f 39.gmln o b t a i n e d from the numerical integration. Fixed Throttle Using Fig 4 T h e p r e s s u r e d r o p across t h e c a k e can be p l o t t e d a g a i n s t t h e r a t e t o y i e l d a m o d i f i e d c h a r a c t e r i s t i c c u r v e . O n F i g 4. t h e c u r v e m a r k e d &P~ r e p r e s e n t s APt. vs q f o r the 2 x 3 p u m p . A h h o u g h the l i m i t i n g t i m e

lines should bc redrawn in accord with E q (19) instead o f Eq (13), little would bc gained for an initial s c r e e n i n g process. D r a w i n g a line ror 75ram thickness leads to an intersection at point D. A crude interpolation yields AP~. o f a b o u t 105-ll0kPa and a time o f a#proximatcly 411-SIImin. A s data involving cake permcabil/ties arc seldom accurate and c h a n g e s in p r o p e r t i e s frequently occur, little would hc gained by e m p l o y i n g a m o r e accurate p r o c e d u r e . A c k n o w l c d K e m ~ . n L 1"he author.'*; w i ~ h I o t h a n k t h e e l ' f l e e o f l]a.~it: E t l c r t ! v ~t~icnc.~.% I'~)r G r : I l l l D ¢ - A S I | 5 - g I E R - I I | 9 4 r ~ w h i c h ha~ e n a b l e d t h t : r n I n c a r r y u n b a s i c r e s e a r c h i n t h e f i e l d o1'

solid-liquid sep:=radon.

Coagulation and Flocculation m

m

~

=

~

~m~ n

Order

m

~

m

Form

m=m a

I

m

APPENDIX M a n y d i f f e r e n t t y p e s o f c o n s t i t u t i v e e q u a t i o n s cap b e e m p l o y e d to r e p r e s e n t e x p e r i m e n t a l data. In as m u c h as p o w e r f u , l c t i o n s have b e e n f a v o u r e d , w e have c h o s e n t h e f o l l o w i n g m o d i f i e d p o w e r f u n c t i o n f o r m u l a s for o u r e x a m p l e s : Ps ~ P.,i P~' ( ~ s ) TM ( t ~ ) ~'' ( ~..~.L) " ' (A 1 ) -~-E, = E = ~ = 0 ~ Ps ~ Psi 0"=c~,

for

Signat-,~e Date

............................................................

.............................................................................

.............................................................................. ......................................................................

(A2)

(AJ)

~t'sk = aBJ = 1

(A41

~..:,..,

1 -- a

AFI~ (1 - nI&FPe-"

~s'-'~ = 1 - n

-1 - - b l A F I & ' "

c e a " - (1 - n) Allen ~'--S"1 - n/an'C ~ K.~.

K-"-~ = ~

1

(&FI;'~ -- b l a n c )

(A51 (AS) (A71

w h e r e A R c = t~Pe/Ps~, T h e s e e q u a t i o n s p r o v i d e r e a s o n a b l y a c c u r a t e r e p r e s e n t a t i o n o v e r p r e s s u r e r a n g e s n o r m a l l y e n c o u n t e r e d in filtration f o r m o d e r a t e l y c o m p r e s s i b l e c a k e s with n less than 0.6 and b less than 0.8. F o r l a r g e r v a l u e s o f the c o m p r e s s i b i l i t y p a r a m e t e r s , the r a n g e o f validity usually d e c r e a s e s substanti~tlly. W h e n n is less t h a n a p p r o x i m a t e l y 0.6 and the p r e s s u r e is not t o o low, Ecls (A5)-{A71 Can be a p p r o i x i m a t e d by Eqs 161-18). For h i g h l y c o m p r e s s i b l e c a k e s with n > 1, Eqs (AS)-(ATI s h o u l d b e r e a r r a n g e d as f o l l o w s : ¢~av (b - 1) In - llA,'l"ln - ' ) ~s"~-= "(n - 1) (,b - l/&,-x;"-') (AS)

K.. 1 K-S- = ~ - 1

Al-li n - l / A R t '-1 (~ ~

-

1 ) an?

lAg) (AIO)

.. ........

...: .........................................................................

...................................................................................... I

52

K=Kt

B a s e d u p o n the local v a l u e s g i v e n in Eqs (A21 a n d (AJ). a v e r a g e values can be d e v e l o p e d as f o l l o w s :

a'a, = (1 - n ) ~i

Job Title

Prc~s L z d . I . o n d , n .

• Gould Pumps. Bull 725:4. S¢:ncc;=Falls. NY 1314P;(Oct 1977).

mmm m m mmE

....................................................................................

Business

Upland~

2. W~tlkcr. W I !. L~vi~. W K. McAdam~. W tl. aml Gillihmd. V. R. "Principles ,ff Chemical 1"ngim:crin~ ". p35h. 2ml cd. McGtaw-Itill Book ('o. New ~nrk (It).'lT) 3. Chcrcmisin'blf. N P. and Azbcl. D S. "Liquid Fillrati(m'. Chaptur 9. Ann Arbor Sclcncc Publishers,Wuburn. MA ( 19~31. 4. "Filler. F M. "The R¢~lcof Porosity in Fihrallun. Part 3: Varinhlc-Pr'cssurc-Variahlc-Ratc Fihratinn'. Al(_'h E l . . 4. 1711( I U5~). 5. Shirato. M. Ara,,aki, T. Mori. R. and Im:d K. 'Slutlics (;n V;triablc Prussur,J-Variablc t~;l|c F~ttnlfion,' X'rtXaK*~ K ~ W , K u . 33. 57b 1191)9), ~. Svart)vsky. L. "Solid-LiquidSeparation'.2nd ctt. p25fi, Buttcrwort ~s Lond(m( 19~1).

Ps = (~JB) ~l=t= (crla) 1"~ = (KJJ) `t"~

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So:parallel1 "l'u¢]mnh)~y'.

w h e r e ~ = n+l]. T h e f o l l o w i n g r e l a t i o n s h o l d :

"Coagulation and Flocculation"

Name

'Sulid/lJtluid

¢,=~j

TO U P L A N D S PRESS LTD., SUITE 28, C E N T R E P O I N T HOUSE, ST GILES HIGH STREET, L O N D O N W C 2 H 8LW, E N G L A N D

FirmlOrganisat|on

REF'ERENC[.~; I. I)urch;e.. Derek.

Eq (A1 } can also be w r i t t e n in the form:

WITH AN EMPHASIS ON WATER AND WASTEWATER TREATMENT m

Greek S y m b o l s S p e c i f i c f l o w r e s i s t a n c e b a s e d on v o l u m e o f solids, m -~, V a l u e o f t~ w h e n Ps <~ P.~, m -~ A v e r a g e value o f ~', m "= ~'a. E x p o n e n t in Eqs (6) a n d (A1), (-) b E x o o n e n t in Eqs (8) a n d (A1), ( - ) ~s Solidosity, v o l u m e f r a c t i o n o f solids, (-) ~si V a l u e o f ~s w h e n P~ .<- P.i, (-) E~.~ A v e r a g e v a l u e o f ~ , (-~ IJ Viscosity, P a . s ~1-[ c APc/p~, {-) ~_~ V o l u m e f r a c t i o n o f s o l i d s in slurry, m~l(-)mZ m= V o l u m e Of s o l i d s / u n i t a r e a in cake, a,

"['he time is next e s t i m a t e d by E q (19):

m

Nomenclature ........ a C o n s t a n t in Eqs (7) and {AJ), d i m e n s i o n s m e a n i n g l e s s B C o n s t a n t in Ecls (6) a n d (AJ), d i m e n s i o n s m e a n i n g t e s ~ e D e f i n e d by Eq (2), v o l u m e o f s o l i d s / u n i t v o l u m e of filtrate (-) F Ratio ~.d4i, s, ( - ) ' J C o n s t a n t in Eqs (81 a n d (AJ), d i m e n s i o n s m e a n i n g l e s s K Permeability, m = K/ Value o f K w h e n P ~ P-~i, m z Kn v A v e r a g e v a t u e o f K, m z L C a k e thickness, m r2_ E x p o n e n t in Eqs (7) and (A1), (-) P Total pressure, Pa o r kPa PPA~ Effective pressure, Pa L o w value o f P= b e l o w w h i c h a', ~'~, a n d K a r e c o n s t a n t , Pa c Pressure d r o p a c r o s s cake P a e l kPa Cl F Itrate f l o w rate/unit area, m~/m ;' • s Rm Medium resistance, I/m t Time, sec v V o l u m e of filtrate/unit area, mJ/m 2

I

""

Use o f Eqs (A8I-(A10) must be a p p r o a c h e d with caut on. E x p e r e n c e in the a p p l i c a t i o n o f t h e s e f o r m u l a s is essential t o t h e i r p r o p e r i n t e r p r e t a tion. U n d e r s o m e c i r c u m s t a n c e s they yield o n l y qualitative results. N e v e r t h e l e s s , the b e h a v i o u r o f c a k e s with n > 1 is m a r k e d l y d i f f e r e n t f r o m that o f less c o m p r e s s i b l e materials. S e l e c t i o n o f p u m p s f o r use with highly c o m p r e s s i b l e c a k e s must b e a p p r o a c h e d cautiously. January/February

1990

Filtration

& Separation