Computer Applications to Chemical Engineering - American Chemical

26 Optimal Design of Batch Ultrafiltration-Diafiltration Process E L M E R H. HSU, STUART BACHER, and CARLOS B. ROSAS

Downloaded by UNIV OF SYDNEY on October 20, 2015 | http://pubs.acs.org Publication Date: May 30, 1980 | doi: 10.1021/bk-1980-0124.ch026

Merck, Sharp, & Dohme Research Laboratories, Rahway, NJ 07065

Ultrafiltration, which uses selective membranes to separate materials on the basis of different molecular sizes, has become a valuable separation tool for a wide variety of industrial processes, particularly in the separation of dispersed colloids or suspended solids. In many cases where a high degree of separation is desired, a batch ultrafiltration process is used because it is the most economical in terms of membrane area. A comprehensive mathematical analysis of batch ultrafiltration coupled with diafiltration is presented. The time cycle of the ultrafiltration-diafiltration has been correlated with the volume initially charged, percent of solute recovered, membrane area and flux. The optimum diafiltration volumes which result in the minimum cycle time or the minimum membrane area were solved for in terms of the operating conditions. For a product recovery of 96 percent, optimum solutions were obtained and are presented graphically via design charts. The design charts plot the optimum diafiltration volume and total time cycle as a function of other operating conditions, i.e., initial volume, recovery, membrane area and flux. For a recovery other than 96 percent, the optimum solution can be obtained using the equations developed in this paper in a similar manner. Introduction Numerous studies relating to the application of ultrafiltration have been presented in the literature. For example, protein ultrafiltration has been studied by Kozinski (1972). Separations of complex aqueous suspensions and organic solutions have been reported by Bhattacharyya (1974, 1975). Industrial applications have been reviewed by Klinkowski (1978). Theoretical aspects of ultrafiltration have been discussed by Michaels (1968), Porter (1972), Shen (1977) and others. Often where a high degree of separation is desired, a

0-8412-0549-3/80/47-124-457$05.00/0 © 1980 American Chemical Society In Computer Applications to Chemical Engineering; Squires, R., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1980.


458

COMPUTER

APPLICATIONS TO

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ENGINEERING

b a t c h u l t r a f i l t r a t i o n p r o c e s s is p r e f e r r e d b e c a u s e it is t h e most e c o n o m i c a l in t e r m s o f membrane a r e a . However, due t o t h e d e c r e a s e in membrane f l u x as t h e s o l i d s c o n c e n t r a t i o n i n c r e a s e s , t h e b a t c h s e p a r a t i o n is n o r m a l l y c a r r i e d o u t in two s t a g e s . F i r s t t h e d i l u t e b a t c h is c o n c e n t r a t e d t o a s p e c i f i c p o i n t . T h e n , w a t e r is added c o n t i n u o u s l y w h i l e t h e f i l t r a t i o n c o n t i n u e s at n e a r l y c o n s t a n t f l u x . T h i s l a t t e r f i l t r a t i o n s t a g e , when w a t e r is a d d e d t o m a i n t a i n a c o n s t a n t f l u x , is r e f e r r e d t o as diafiltration. Proper c h o i c e o f the d i a f i l t r a t i o n s t a r t i n g time c a n m i n i m i z e t h e r e q u i r e d membrane a r e a , w h i c h is o f t e n t h e m a j o r p a r t o f t h e c a p i t a l c o s t in an u l t r a f i l t r a t i o n p r o c e s s . Due t o t h e c o m p l i c a t i o n o f d i a f i l t r a t i o n , d e t e r m i n a t i o n o f t h e optimum u l t r a f i l t r a t i o n c y c l e n o r m a l l y r e q u i r e s t i m e c o n s u m i n g e x p e r i m e n t a l work o r t e d i o u s c a l c u l a t i o n s . I n this p a p e r , c o m p l e t e m a t h e m a t i c a l f o r m u l a t i o n s f o r c o r r e l a t i n g the time c y c l e s w i t h o t h e r o p e r a t i n g c o n d i t i o n s are presented. The optimum d i a f i l t r a t i o n c y c l e ( i n terms o f v o l u m e f r a c t i o n ) , and t h e t o t a l c y c l e t i m e a r e s o l v e d as f u n c t i o n s o f membrane a r e a , f l u x , i n i t i a l v o l u m e and r e c o v e r y . Convenient c h a r t s , w h i c h c a n be u s e d as a g u i d e in d e s i g n i n g o r m o d i f y i n g an u l t r a f i l t r a t i o n p r o c e s s , a r e p r o v i d e d . Mathematical

Formulation

The s c h e m a t i c o f a t y p i c a l b a t c h u l t r a f i l t r a t i o n p r o c e s s u s e d f o r s e p a r a t i n g s u s p e n d e d s o l i d s is shown in F i g u r e 1. The o p e r a t i n g t a n k is c h a r g e d i n i t i a l l y w i t h a f i x e d v o l u m e of s l u r r y . T h e n , t h e s l u r r y is c i r c u l a t e d c o n t i n u o u s l y t h r o u g h t h e membrane at a h i g h f l o w r a t e . A h i g h degree o f t u r b u l e n c e is m a i n t a i n e d so t h a t t h e c o n c e n t r a t i o n p o l a r i z a t i o n f i l m t h i c k n e s s on t h e membrane s u r f a c e is m i n i m i z e d and t h e h i g h e s t p o s s i b l e f l u x is a t t a i n e d , ( K l i n k o w s k i , 1 9 7 8 ) . As t h e p e r m e a t e is c o n t i n u o u s l y removed, t h e s l u r r y v o l u m e in t h e o p e r a t i n g tank d e c r e a s e s . Thus, the s o l i d c o n c e n t r a t i o n i n c r e a s e s , and t h e f l u x d r o p s a c c o r d i n g l y . To a v o i d h a v i n g a f l u x t o o low t o be p r a c t i c a l at v e r y low o p e r a t i n g v o l u m e , d i a f i l t r a t i o n is a d o p t e d t o w a r d t h e l a t t e r s t a g e o f t h e filtration. D u r i n g the d i a f i l t r a t i o n phase, the r a t e o f water added is k e p t e q u a l t o t h e f l u x so t h a t t h e s o l i d s c o n c e n t r a t i o n and, t h u s , f l u x c a n be m a i n t a i n e d n e a r l y c o n s t a n t . A m a t e r i a l b a l a n c e on during d i a f i l t r a t i o n gives

the the

s o l u t i o n in t h e o p e r a t i n g following equation

- V d C = JrjACdt

(1)

D

Where V C Jj) A t D

= = = = =

volume d u r i n g the d i a f i l t r a t i o n Solute concentration flux during diaf i l t r a t i o n membrane a r e a time

tank

stage

In Computer Applications to Chemical Engineering; Squires, R., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1980.


26.

Batch

Hsu E T A L .

Ultrafiltration-Diafiltration

459

RETURN BLEED

OPERATING TANK PERMEATE

MEMBRANE

1

• Q FEED PUMP Figure 1.

•Q—

RECIRCULATION PUMP

Flow diagram of batch ultrafiltration process


C O M P U T E R APPLICATIONS T O C H E M I C A L ENGINEERING

460

I n t e g r a t i o n of Eq. (1) from the beginning o f the d i a f i l t r a t i o n phase, Tu, to the end, T, gives C

=

f

C

EXP Γ A Jp (Tu - Τ ) η IVD

0

() 2

J

=

Where C i n i t i a l solute c o n c e n t r a t i o n Cf = f i n a l s o l u t e c o n c e n t r a t i o n Q

The

f r a c t i o n recovery

o f the s o l u t e , R, is defined as

R = 1.0 - f v P CoVo


c

(3)

Where Vo = i n i t i a l volume From Equations (2) and (3), the t o t a l time c y c l e , T, can be solved by e l i m i n a t i n g Cf and Co. Τ « Tu

+

^ A J

Γ

in

L

D

By l e t t i n g Κ = AJp/Vo, and U can be r e w r i t t e n as T = Tu

3

Γ

h

Κ

υ

L (1 / Ί

η

VD

D

(

4

)

J

(1 - R) Vo

= V /Vo, the above equation D

Ρ ,Ί - R) J

(5)

From i n s p e c t i o n o f Equation (5), it can be seen that the t o t a l time c y c l e is the sum of u l t r a f i l t r a t i o n and the d i a f i l t r a t i o n c y c l e s with the d i a f i l t r a t i o n c y c l e given by the second term on the right-hand s i d e o f the equation. During the u l t r a f i l t r a t i o n phase, the d i f f e r e n t i a l volume change in the operating tank can be r e l a t e d to the membrane f l u x by - dV = J A dt

(6)

The membrane f l u x , J , is in general a l o g a r i t h m i c f u n c t i o n of the suspended s o l i d s c o n c e n t r a t i o n in the s l u r r y (Michaels, 1968) J = m In f So

+

b

(7)

The slope, m, and the i n t e r c e p t , b, are constant f o r a given u l t r a f i l t r a t i o n process. Since the product of s o l i d s c o n c e n t r a t i o n and t o t a l volume in a given batch is always a constant, the f l u x can a l s o be expressed as J = m In ψ

one

+

b

(8)

S u b s t i t u t i n g the above r e l a t i o n s h i p f o r J in Equation ( 6 ) , obtains


26.

HSU E T

Batch

AL.

- g at

=


m A ln ^ ν

+

461

bA

(9)

There are four parameters (m, b, A, and Vo) in the above equation. L e t t i n g U = V/Vo, Ρ = mA/Vo, and Q = bA/Vo, Equation (9) is reduced to a two-parameter equation

*

" Έ

p

Φ

l n

+

Q


By s u b s t i t u t i n g X = Ρ l n ( ^ ) + Q and ^ Equation (10) can be transformed to

(

e -Q

/ P )

Ρ dt = !

d

= - P ^

(11)

X

I n t e g r a t i o n between the beginning of the o p e r a t i o n (t = 0; V = Vo) to the end of the u l t r a f i l t r a t i o n phase (t = Tu; V = Vp) y i e l d s the expression f o r the u l t r a f i l t r a t i o n time c y c l e . 1 Tu

= -

. +

e

(Q/P)fr w

/

r

Ί[ΐη

/

[Ρ

[ P l n (Vo/Vp) + Q ]

In

,

χ

(VO/VD) +

,

Ρ In (Vo/Vn) + Q ψ- * 3

Q]

[ P l n (Vo/Vp) + Q ]

2

TT.—T~P2

3-3!·

—

3

*

1

.

P3

+

η . . . .J

(12)

Since the i n i t i a l volume, recovery, membrane area and f l u x parameters (m, b) are a l l constant and known or s p e c i f i e d , the t o t a l batch c y c l e time is a f u n c t i o n of the r e l a t i v e d i a f i l t r a t i o n volume (Vo/Vd) o n l y . The optimum can be determined by d i f f e r e n t i a t i n g Equation (5) with respect to Up and s e t t i n g the r e s u l t to zero, i . e . |T 3Up OR

=

â

u

I

+

9 Up

|Tp . 3 up

(

Di + D2 = 0

1

m

Dl

7

1 D2 = ^

(

0

/

3

)

(14)

p

_

)

fJL

e

LiCUp

L·

+

+

Up

Ρ In [(1-R)/Up] - ^2

Κ = Ρ ln ^UP

1

+

_ 2Kp(P/UD)

2-2!

+

·

In [U-R)/UpJ

3K

2 D

(P/UD) _

3-3!

η *

. p3

1 + £

Q


J

(

1

5

)

(

1

6

)

(17)

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E q u a t i o n ( 1 4 ) is an i m p l i c i t a l g e b r a i c e q u a t i o n o f t h e optimum r e l a t i v e d i a f i l t r a t i o n v o l u m e , U D . I t c a n be s o l v e d n u m e r i c a l l y by any one o f a number o f m e t h o d s , e. g. Newton, Raphson T e c h n i q u e , ( L a p i d u s , 1 9 6 2 ) . Once t h e v a l u e o f U D is d e t e r m i n e d , t h e optimum t i m e c y c l e s o f t h e u l t r a f i l t r a t i o n and d i a f i l t r a t i o n s t a g e s , Tu and To, c a n be c a l c u l a t e d r e a d i l y f r o m E q s . ( 1 2 ) and ( 5 ) .


Use

of Design

Charts

I t has b e e n shown t h a t t h e r e l a t i v e d i a f i l t r a t i o n v o l u m e and t h e t o t a l t i m e c y c l e c a n be s o l v e d f o r in terms o f t h r e e p a r a m e t e r s , P, Q, and R ( E q s . ( 1 4 ) and ( 5 ) ) . T h u s , at a g i v e n r e c o v e r y , R, t h e v a l u e s o f U D and Τ c a n be o b t a i n e d f o r v a r i o u s v a l u e s o f Ρ and Q. C o n s i d e r i n g R as a p a r a m e t e r , p l o t s o f U D and Τ as f u n c t i o n s o f Ρ and Q c a n be made. The c u r v e s f o r b a t c h u l t r a f i l t r a t i o n w i t h i n t h e f o l l o w i n g o p e r a t i n g r a n g e s (P = 0.02 t o - 0.38; Q = 0.1 t o 0.60; R = 0.96) a r e p r e s e n t e d in this p a p e r ( F i g u r e s 2 and 3 ) . The u s e o f t h e c h a r t s is s t r a i g h t f o r w a r d . For each r e c o v e r y , t h e r e a r e two c o r r e s p o n d i n g c h a r t s . One determines t h e optimum t i m e c y c l e and t h e o t h e r d e t e r m i n e s t h e optimum d i a f i l t r a t i o n volume. F o r t h e c a s e where t h e i n i t i a l v o l u m e , membrane f l u x , d e s i r e d r e c o v e r y , and t h e t i m e c y c l e a r e s p e c i f i e d o r known, t h e r e q u i r e d membrane a r e a c a n be d e t e r m i n e d f r o m t h e c o r r e s p o n d i n g Time C y c l e C h a r t . The p r o c e d u r e is t o f i r s t c a l c u l a t e Ρ and Q b a s e d on an assumed a r e a . Then, the t i m e c y c l e is f o u n d f r o m t h e c h a r t . F i n a l l y , t h e a r e a is a d j u s t e d u n t i l the time c y c l e r e a d from the c h a r t matches the s p e c i f i e d time c y c l e . Once t h e a r e a is d e t e r m i n e d , t h e optimum r e l a t i v e d i a f i l t r a t i o n v o l u m e c a n be f o u n d f r o m t h e c o r r e s p o n d i n g r e l a t i v e d i a f i l t r a t i o n Volume C h a r t . F o r t h e c a s e when t h e membrane a r e a and r e c o v e r y a r e known, t h e optimum c y c l e t i m e and t h e r e l a t i v e d i a f i l t r a t i o n v o l u m e , f o r any amount o f i n i t i a l c h a r g e w i t h any f l u x r a t e c a n be r e a d d i r e c t l y from the c h a r t . Thus, the c h a r t s not o n l y a s s i s t the d e s i g n o f new p r o c e s s e s , t h e y a l s o p r o v i d e a q u i c k q u i d e t o t h e e x i s t i n g p l a n t in a d j u s t i n g t h e p r o p o r t i o n o f t h e d i a f i l t r a t i o n c y c l e when o p e r a t i n g c o n d i t i o n s a r e c h a n g e d o r t h e membrane f l u x is a l t e r e d due t o l o n g - t e r m f o u l i n g o r d e t e r i o r a t i o n . They e l i m i n a t e t h e n e e d f o r t e d i o u s c a l c u l a t i o n and m i n i m i z e t h e amount o f e x p e r i m e n t a l work r e q u i r e d t o p r o v i d e an u l t r a f i l t r a t i o n process design. Examples Use o f t h e d e s i g n c h a r t s examples : Case

1.

Determination

is

illustrated

o f Membrane A r e a .

by

the

following

Assume a

batch,


two

In Computer Applications to Chemical Engineering; Squires, R., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1980. Figure 2.

Volume chart (R =

0.96)


1

ο

I

δ'

8*

«s.

2

4

CI

SS

Ci

«s.

to

r

M H >

d

Χ

to

COMPUTER

464

APPLICATIONS

TO

CHEMICAL

ENGINEERING

oo


CO

(·ΗΗ)

Q0Id3d

ONIlud3dO


26.

Batch

Hsu E T A L .


11,500 g a l l o n s in v o l u m e , t o d i a f i l t r a t i o n technique. If and t h e f l u x c u r v e f o l l o w s J f t , t h e n , t h e membrane a r e a follows : 2

1)

An a r b i t r a r y

2)

Ρ and Q a r e c a l c u l a t e d


Ρ

4)

A = 2300 f t

2

.

= mA/Vo = - 0.064

Τ is o b t a i n e d Since more

5)

be p r o c e s s e d by t h e u l t r a f i l t r a t i o n 96% is t o be r e c o v e r e d in 8 h o u r s , = - 0.32 l n ( V o / V ) + 0.88 g a l / h r r e q u i r e d c a n be d e t e r m i n e d a s

a r e a is p i c k e d ,

Q = bA/Vo = 3)

465

0.176 using

Τ is l o n g e r than

2300 f t

F i g u r e 3 , Τ = 11 h o u r s

than 2

8 hours,

the area

( P i c k A = 4600 f t

2

r e q u i r e d must be

) .

Ρ and Q a r e r e c a l c u l a t e d . Ρ

= -

Q =

0.128 0.352

6)

Τ is o b t a i n e d

7)

Now,

using

Figure 3

Τ is s h o r t e r t h a n

Τ = 5.8

8 hours,

hours

A is d e c r e a s e d .

(Pick A =

3000). 8)

Again, Ρ

= -0.0835

Q = 9)

Ρ and Q a r e c a l c u l a t e d .

0.23

Τ is o b t a i n e d

using

Figure

3

Τ = 8 hours 2

T h e r e f o r e , t h e r e q u i r e d membrane a r e a is 3000 f t . Now, w i t h Ρ = -0.0835 and Q = 0.23, u s i n g t h e v o l u m e c h a r t , ( F i g u r e 2 ) , t h e optimum d i a f i l t r a t i o n v o l u m e is f o u n d t o be 17%. C a s e I I ; D e t e r m i n a t i o n o f Optimum C y c l e Time & D i a f i l t r a Volume. I f t h e same p r o c e s s e q u i p m e n t is u s e d t o p r o c e s s t h e same amount o f b r o t h (Vo = 11,500 g a l ; A = 3,000 f t ) , b u t t h e f l u x h a s d r o p p e d 20% ( a s a r e s u l t o f f o u l i n g ) , t h e n f o r t h e same r e c o v e r y ( 9 6 % ) , t h e t o t a l p r o c e s s i n g t i m e and t h e optimum d i a f i l t r a t i o n v o l u m e c a n be f o u n d d i r e c t l y f r o m F i g u r e s 3 and 2. 2


466

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Given:

R = 0.96 Vo = 11,500 m = -0.32 χ 0.8 = -0.256 b = 0.88 χ 0.8 = 0.704 Calculated: Ρ = mA/Vo = -0.0668 Q = bA/Vo = 0.184 Found f r o m F i g . 3: Τ = 10.3 h o u r s f r o m F i g . 2: V D / V O =0.18


Conclusions The optimum t i m e c y c l e and t h e r e l a t i v e d i a f i l t r a t i o n v o l u m e in t h e u l t r a f i l t r a t i o n - d i a f i l t r a t i o n p r o c e s s c a n be e x p r e s s e d as a f u n c t i o n o f t h r e e v a r i a b l e s , P, Q, and R. Ρ and Q a r e s i m p l e f u n c t i o n s o f t h e i n i t i a l v o l u m e , membrane a r e a , and f l u x (P = mA/Vo, Q = b A / V o ) , and R is t h e s o l u t e r e c o v e r y . From t h e s e , t h e t i m e c y c l e and r e l a t i v e d i a f i l t r a t i o n v o l u m e ( V D / V O ) c a n be s o l v e d at v a r i o u s v a l u e s o f m, b, Vo, A, and R (m and b a r e r e s p e c t i v e l y t h e s l o p e and i n t e r c e p t o f t h e f l u x , J = m I n Vo/V + b ) . A t a f i x e d r e c o v e r y , t h e optimum t i m e c y c l e and t h e r e l a t i v e d i a f i l t r a t i o n v o l u m e become f u n c t i o n s o f o n l y two v a r i a b l e s Ρ and Q. T h u s , t h e optimum o p e r a t i n g c o n d i t i o n c a n be s i m p l y p l o t t e d as f u n c t i o n o f Ρ and Q. These p l o t s , p r o v i d i n g c o n v e n i e n t and s u f f i c i e n t i n f o r m a t i o n , c a n be u s e d as a g u i d e in t h e d e s i g n and o p e r a t i o n o f t h e u l t r a f i l t r a tion process. The d e s i g n c h a r t s and t h e e x a m p l e s p r o v i d e d in this p a p e r i l l u s t r a t e t h e s i m p l e p r o c e d u r e o f s o l v i n g a common u l t r a f i l t r a t i o n problem. I n g e n e r a l , when P, Q and R f a l l b e y o n d t h e c o v e r e d r a n g e s , a d d i t i o n a l c h a r t s c a n be r e a d i l y p r e p a r e d by s o l v i n g t h e i m p l i c i t e q u a t i o n s p r e s e n t e d in this p a p e r .

Nomenclature A b c Cf Co Di

= = = = = =

Membrane a r e a Intercept of f l u x curve Solute concentration Final solute concentration I n i t i a l solute concentration D e r i v i t i v e o f u l t r a f i l t r a t i o n time c y c l e w i t h r e s p e c t to UD D e r i v i t i v e o f d i a f i l t r a t i o n time c y c l e w i t h r e s p e c t to

D2

-

J JD

= "

Flux Flux during

Κ

=

JDA/VO

m Ρ Q R

= = = =

Slope of f l u x curve mA/Vo bA/Vo F r a c t i o n recovery of

UD

diafiltration

phase

solute


26.


S So t

Batch

HSU E T AL.

= = =


467

Suspended s o l i d c o n c e n t r a t i o n I n i t i a l suspended s o l i d c o n c e n t r a t i o n Time

Τ

=

Total

TD Tu U UD V

= = = = =

Time c y c l e o f d i a f i l t r a t i o n p h a s e Time c y c l e o f u l t r a f i l t r a t i o n p h a s e Volume f r a c t i o n r e m a i n e d ( V / V o ) Volume f r a c t i o n r e m a i n e d in d i a f i l t r a t i o n p h a s e ( V D / V O ) Solid free liquid v o l u m e in o p e r a t i n g t a n k

VD Vo X

= = =

O p e r a t i n g v o l u m e in d i a f i l t r a t i o n Initial liquid volume JA/Vo

Literature

time

cycle

phase

Cited

Bhattacharyya, Dibakar, et. al., AIChE J., 20, Bhattacharyya, Dibakar, et. al., AIChE J., 21, Klinkowski, P.R., Chemical Engineering, May 8, Kozinski, A.A., Lightfoot, E.N., AIChE J., 18, Lapidus, L., "Digital Computation for Chemical

1206 (1974) 1057 (1975) 165 (1978) 1030 (1972) Engineers",

Chapter 6, McGraw-Hill, New York, 1962. Michaels, A.S., Chem. Eng. Progr. 64 (12), 31 (1968) Porter, M.C., Ind. Eng. Chem., Prod. Res. Dev. 11, 234 (1972) Shen, Joseph J.S., Probstein, Ronald F., Ind. Eng. Chem., Fundam.

RECEIVED

16, 459

(1977)

November 5, 1979.


Computer Applications to Chemical Engineering - American Chemical

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