krd important

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P e r g a m o n

Chemical En ineering Science, Vol. 50, No. 2, pp. 223 230, 1995

Copyright 1995 Elsevier Science Ltd

Printed in Gre at Britain. All rights reserved

0009 2509/95 $9.50 + 0.00

0 0 0 9 - 2 5 0 9 9 4 ) 0 0 2 3 0 - 4

R E S I D E N C E T I M E D I S T R I B U T I O N F O R U N S T E A D Y - S T A T E

S Y S T E M S

J. FERN.ANDEZ-SEMPERE,* R. FONT-MONTESINOS and O. ESPEJO-ALCARAZ

Departamento de Ingenieria Quimica, Universidad de Alicante, Apartado 99, Alicante, Espafia

( R e c e i v ed 1 M a r c h 1994;a c c e p t e d i n r e v is e d f o r m 4 A u g u s t 1994)

Abstract--The concept of residence time distribution (RTD) can also be applied to incompressible luids in

closed-closed systems under nonsteady conditions, when the residence time distribution expressed as

a function of a residence time, defined in this paper, is independent of the volume and/or the flow rate. In

this paper, an analysis of the hydrodynamics of the plug flow reactor (PFR), the continuous stirred tank

reactor (CSTR), the plug flow reactor with axial dispersion (DFR) and a series of n-CSTR under

unsteady-state conditions is made. A generalized residence time distribution is proposed for analyzing the

behavior of the system. The proposed RTD is applied to the experimental data obtained when a tracer is

introduced as a pulse in a sewage system. Three runs, with different outlet flow ranges, were carried out. By

means of the generalized RTD, a disperse flow reactor model for correlating the experimental data was

proposed. In this way, the variation deduced in the tracer outlet concentration can be explained, despite the

fact that the outlet flow range is different from one run to another.

I N T R O D U C T I O N

The use of residence time distrib ution (RTD) curves to

characterize the behavior of reaction systems, espe-

cially continuous reactors at steady state, is well

know n and accepted. A large amount of experimental

infor mation is available in some basic books (Leven-

spiel, 1962; Froment an d Bischoff, 1979; Smith, 1981;

Denbigh and Turner, 1984; Fogler, 1992; Westerterp

e t a l . , 1993). The measuremen t of RTD is based on the

injection of a tracer material in the system and sub-

sequent determination of the tracer concentration in

the fluid leaving the system. Three different methods

are used: (a) injection of the tracer in a very short t ime

interval at the ent rance of the system (pulse injection);

(b) introduction of a concentration change in the

form of a step function and (c) intr oduct ion of a peri-

odic concent ration fluctuati on in the inflow. F rom the

information obtained from any of these methods, the

behavior of a certain element of fluid can be known.

This is normally when steady-state conditions are

considered: the inlet flow to the system is always the

same and the system volume remains constant. This is

usually the case when a cont inuous reactor is studied.

Nevertheless, little information has been obtained

for systems under unsteady-state conditions. Na uma n

(1969) studied the RTD for a stirred tank reactor. Fan

e t a l . (1979) proposed a stochastic model of the un-

steady-state age distribution in a flow system.

Schwartz (1979) applied the basic concepts of turn-

over time, mean age and mean transit time to the

atmospheric SO2 and sulfate aerosol. Vaccari

e t a l .

(1985) considered the growth of micro-organisms in

unsteady-state activated sludge system. Calu and

Lameloise (1986) studied the RTD in systems of vari-

*Author to whom correspondence should be addressed.

223

able volumic mass flow and applied it to evaporators

in sugar plants. Dickens e t a l . (1989) studied the RTD

for unsteady flows in a baffled tube.

When env ironmental problems are studied (con-

taminating spills is a sewage system, lagoon-purifying

plants, etc.), in many occasions the flow is not con-

stant and in some cases the volume is variable. In this

case, two phenomena can affect the model flow: (a) the

changes due to the inlet flow and/or to the system

volume, and (b) the changes due to the nonsteady

conditions. Therefore, when a tracer injection is used

in this type of system, the RTD will be modified by

random flow and/or volume changes. Nevertheless, in

spite of these phenomena, the system flow can be

approximately characterized as indicated in this pa-

per, when a reduced time (defined in the following

sections) is independe nt of the volume and /or the flow

rate.

RTD IN CLOSED ~2LOSED SYSTEMS

The analysis presented in this paper is applied to

closed-closed systems with incompressible fluids.

Nevertheless, it is possible that some conclusions can

be applied in other circumstances also. Two different

situations can occur when the inlet flow and the outlet

flow vary:

--The inlet flow is the same as the outlet flow, for

any time, although the flow varies with time.

Therefore, the volume of the system is constant.

--As the inlet flow is different from the outlet flow,

the volume of the system also changes.

These two cases are analyzed in this paper. In both

cases, and for a par ticu lar mass of fluid, residence time

distribution function (E) similar to that used when the


2/8

224

f l o w i s c o n s t a n t a n d t h e s y s t e m i s i n s t e a d y s t a t e i s

d e f i n e d .

C o n s i d e r a s y s t e m a s i n F i g . l ( a ), w i t h v a r i a b l e i n l e t

a n d o u t l e t f l o w s w h i c h a r e n o t n e c e s s a r i l y e q u a l . A t

a c e r t a i n t i m e , a f l o w o f fl u i d e n t e r s t h e s y s t e m . B e -

t w e e n t a n d t + d r , t h e f r a c t i o n o f t h i s f l o w t h a t l e a v e s

t h e s y s t e m c a n b e d e f i n e d b y E d t . L o g i c a l l y , t h e

d i s t r i b u t i o n f u n c t i o n E w i ll b e d i f fe r e n t i n o t h e r s i t u -

a t i o n s . I n s o m e c a s e s , h o w e v e r , i t c o u l d b e p o s s i b l e t o

k n o w E a n d c h a r a c t e r i z e t h e s y s t e m ( f o r a s y s t e m o f

c o n s t a n t f l o w a n d u n d e r s t e a d y s t a te , t h e f u n c t i o n E i s

t h e s a m e f o r a l l t h e f r a c t i o n s o f f l u i d e l e m e n t s f l o w i n g

t h r o u g h t h e s y s t e m ) .

F u n c t i o n E c a n b e o b t a i n e d b y m e a n s o f a p u ls e

t e s t: A t t = 0 , a c e r t a i n a m o u n t o f t r a c e r M i s i n s e r t e d

i n t o t h e s y s te m a n d t h e t r a c e r c o n c e n t r a t i o n i n t h e

o u t l e t s e c t i o n is t h e n a n a l y z e d . F i g u r e l ( b ) sh o w s t h e

v a r i a t i o n o f t h e t r a c er c o n c e n t r a t i o n C a n d t h e i n l e t

a n d o u t l e t f l o w v s t i m e . T h e t r a c e r i s d i s t r i b u t e d

c o m p l e t e l y i n t o t h e i n l e t f l u i d , s o t h e o u t l e t f l o w

c o n c e n t r a t i o n i s a n i n d i c a t i o n o f t h e p r e s e n c e o f fl u i d

e l e m e n t s s t a i n e d w i t h t h e t r a c e r , a s o c c u r s i n s t e a d y

s ta t e .

T h e t r a c e r t o t a l a m o u n t ( M ) c a n b e c a l c u l a t e d f r o m

t h e e x p r e s s i o n :

M = d M = C d V = C Q o d t

(1)

0 0 0

M b e i n g t h e s u m o f a l l t h e t r a c e r m a s s e l e m e n t s

p r e s e n t a t e a c h m o m e n t i n t h e f lu i d l e a v i n g t h e s ys -

t e m , C th e t r a c er c o n c e n t r a t i o n a n d Qo t h e o u t l e t f l o w .

E q u a t i o n ( 1 ) c a n b e w r i t t e n a s

f

o~ ( CQ o/M ) =

(2)

t

1

0

w h e r e

( C Q o /M ) d t = A M / M .

(3)

T h e e x p r e s s i o n

C Q o / M

r e p r e s e n t s t h e f r a c t i o n o f

f l u id e l e m e n t s r e m a i n i n g i n t h e s y s te m a t a t i m e b e -

t w e e n t a n d t + d t , a n d s o

( C Q o / M ) d t = E d t

(4)

J . F E R N A N D E Z - S E M P E R E

et a l

a n d

C Q o / M = E .

(5)

T h e m e a n r e s i d e n c e t i m e f o r t h e m a s s o f fl u i d c o n -

s i d e r e d c a n b e c a l c u l a t e d b y

f

oo E t dt

f = o

o~ E t dt.

6 )

0

~ E d t

0

T h e t w o c a se s p r e v i o u s l y m e n t i o n e d a r e a n a l y z e d

as fo l lows .

S y s t e m s w i t h c o n s ta n t v o l u m e

I n t h e c a s e o f s y s t em s w i t h v a r i a b l e f l o w b u t w i t h

a c o n s t a n t s y s t e m v o l u m e ( i n l e t f l o w = o u t l e t f lo w ) ,

t

t h e v a l u e s o f

E / Q o

c a n b e p l o t t e d v s S o

Q o d t .

T h e

f o l l o w i n g c a n b e d e d u c e d :

( E /Q o) d [ ; o Q o d t ] = ( E / Q o ) Q o d t = E d t

(7)

a n d t h e r e f o r e

r e p r e s e n t s t h e f r a c t i o n o f f l u i d e l e m e n t s t h a t e n t e r s t h e

r e a c t o r a t t = 0 a n d l e a v es t h e r e a c t o r b e t w e e n t a n d

t + d t .

O n t h e o t h e r h a n d , f r o m e q . ( 5 )

E /Q o = C / M .

(8)

N o w c o n s i d e r t h e f o l l o w i n g ca se s.

P l u # f l o w t h r o u g h a s y s t e m o f c o n s ta n t v o l u m e

(V)

w i t h v a r i a b le f l o w

(Q o ). T h e d i s t r i b u t i o n f u n c t i o n w i l l

a l w a y s b e t h e s a m e : a D i r a c d e l t a f u n c t i o n t h a t a p -

p e a r s w h e n t h e p u l s e i n j e c t i o n is i n t h e e x i t s e c t i o n .

T h i s o c c u r s , t a k i n g i n t o a c c o u n t t h a t t h e f l u i d i s

i n c o m p r e s s i b l e , w h e n t h e f l u i d v o l u m e t h a t h a s l e ft t h e

s y s t e m f r o m t h e i n l e t o f t h e p u l s e i n j e c t i o n i s th e s a m e

a s t h e s y s t e m v o l u m e .

L

a ) . variable volume

V

b

Q i

Qo

t=O t

Fig. 1. (a) System with inlet flow, outlet flow and volume

changing with t ime. (b) Varia t ion of the out le t f low Qo and

the t racer co ncentra t ion with t ime.

I d e a l s t ir r e d t a n k r e a c t o r o f c o n s t a n t v o l u m e

(V)

w i t h

v a r i a b le f l o w

( Q o) . I f a p u l s e i n j e c t i o n i s i n t r o d u c e d i n

t h e s y s t e m w i t h a t r a c e r a m o u n t , M , a n d t h e r e f o r e

w i t h a n i n i t ia l c o n c e n t r a t i o n :

C o = M / V

(9)

t h e t r a c e r b a l a n c e w i ll b e

Q oC = - d ( V C ) / d t = - V d C / d t

(10)

a n d , t h e r e f o r e ;

- d C / C = ( Q o /V ) d t.

I n t e g r a t i n g b e t w e e n C = C o f o r t = 0 a n d C = C

f o r t = t , t h e f o l l o w i n g c a n b e o b t a i n e d :

[ f o l

= C o e x p - ( l / V )

Q o d t .

(11)


3/8

Residence time distribution for unsteady-state systems

Fr om eqs (5) , (9) and (11)

E = C Qo / M = C o ex p [ ( - 1 / V ) f o ' Qo d t ] Qo / M

= ( Q o / V ) e x p [ ( - l /V ) fo Q o d t(12)

and , the re fore ,

E/Qo = ( l / V ) e x p [ ( - 1 / V ) f o ' Q o d t ] . (13)

T h i s e x p r e s s i o n i s e q u i v a l e n t t o t h a t p r e s e n t e d b y

Na u m a n ( 1 9 6 9 ) .

I f t h e s y s t e m v o l u m e ( V ) i s f ix e d a n d k n o w n , E/Qo

d e p e n d s o n l y o n t h e i n t e g r a l v a lu e a n d i s n o t a f f e c te d

b y t h e v a r i a t i o n o f Qo with t ime.

Cascade of N equal ideally s tirred tanks o f constant

volume

(V)

with variable f low

(Qo) . The previous case

c a n b e e x t e n d e d t o a c a s c a d e o f N t a n k s . T h e r e s u l ti n g

e q u a t i o n w o u l d b e

( E / Q o ) = ( N / V ) ( N / V ) Q o d t [ I / ( N - 1 ) ]

x e x p [ ( - - N / V ) f f Q o d t] (14)

and , aga in , E/Qo d e p e n d s o n l y o n t h e i n t e g r a l v a l u e

S; Qo dt.

Ax ially dispersed plug f low reactor of constant vol-

ume (V) with variable f low (Qo). In th i s case , a s sumin g

a c o n s t a n t r e a c t o r s e c t i o n , t h e m a s s b a l a n c e c a n b e

e x p r e s s e d b y

6 C / 6 t = D ~ 2 C / ~ x 2 - - U

(~C/(~X

( 1 5 )

w h e r e

u = (Qo/S) = Qo/(V/L) = QoL/V (16)

F r o m e q s ( 15 ) a n d ( 16 )

: o

C/~ Qo d t = (D/Qo) (62 C/6x a) - {L /V) (6C/6x ) .

(17)

W h e n D v a r i e s a p p r o x i m a t e l y l i n e a r l y w i t h Qo,

D/Qo c a n b e c o n s i d e r e d c o n s t a n t a n d t h e r e f o r e t h e

di s t r ibu t ion func t ion wi l l on ly depend on So Qo dt.

Dimensionless eq uations

W h e n t h e f lo w i s c o n s t a n t , t h e R T D f u n c t i o n c a n b e

expres sed in a d ime ns ionles s w ay (E0) as a func t ion of

0 = t/F, (18)

the d imens ionles s t ime . S imi la r ly , when the f low i s

v a r i ab l e , d i m e n s i o n l e ss e q u a t i o n s c a n b e o b t a i n e d f o r

t h e v a r i a t i o n

E/Qo = f [ fot Q d tl (19)

225

I n t h i s c a s e t h e d i m e n s i o n l e s s e q u a t i o n w o u l d b e

[ f o l

V /Qo = f ( l / V ) Q o d t (20)

w h e r e

f 0

o = E V / Q ,

a n d

O = ( I / V ) Q o d t

( 2 1 )

a n d t h e r e f o r e

Eo = f ( 0 ) . ( 2 2 )

According to th i s de f in i t ion , the d i s t r ibu t ion func-

t ion for the cases cons ide red previous ly wi l l be the

fo l lowing:

Plug f low reactor. In th i s case , the input s igna l

appears in the out l e t w i th a d imens ionles s t ime de lay

equa l to 1 .

Continuous ideally stirred tank reactor. F r o m a m a ss

ba lance , eq . (23) can be obta in ed:

Eo = exp ( - 0 ) . (23)

Cascade of N equal ideally mixed tank reactors.

F r o m a m a s s b a l a n c e , t h e f o ll o w i n g e q u a t i o n is o b -

t a i n e d i f t h e n u m b e r o f t a n k s ( N) i s c o n s t a n t :

Eo = [N(NO ) N 1/(N -- 1 ) ] e x p ( - - N 0 ) . ( 2 4 )

Axia lly dispersed plug f lo w reactor. In this case, eq.

(17) can be wr i t t en as

6 C / 6 0 = 6 C / 6 I ( 1 / V ) f ~ Q o d t ]

= (D V/Qo L 2) (32 C / 6 z 2 ) - 6 C / 6 z (25)

where z i s a d imens ionles s l ength ( z = x /L) a n d t h e

di spers ion module i s

D V/Qo L 2 = D /uL.

I f th i s mo dule i s cons tan t , i . e . t he ra t io D/Qo is

c o n s t a n t , a ll t h e e q u a t i o n s p r o p o s e d i n t h e R e f e r e n ce s

remain val id.

Systems with variable volume

In th i s case , a d i s t r ibu t ion

c a n b e d e f i n ed , w h e r e V i s t h e r e a c t o r v o l u m e a n d Qo

i s the out le t f low.

T h e e x p r e s s i o n (EV/Qo) d [So (Qo /V)dt ] w h i c h

e q u a l s E dr, represent s the f rac t ion of f lu id e le -

me nt s l eaving the sys tem be tween t and t + d t , o r be -

tween the d imens ionles s t imes [ ' .(Qo/V) dt a n d

So (Qo/V) dt + (Qo/V)dt

The expression'U---

EV/Qo

equa l s the d imens ionles s res idence t ime d i s t r ibu t ion

t

E* and So (Qo/V)d t represent s the d imens ionles s

t ime 0" .

T h e d i s t r i b u t i o n f u n c t i o n d e f i n e d i n t h i s w a y h a s

the adv anta ge tha t , i n the fo l low ing cases , it dep end s

o n l y o n s o m e p a r a m e t e r s a n d c a n t h e r e f o r e b e u se fu l


4/8

226

J . FERNANDEZ-SEMPERE

et al .

for studying the hydrodynamic behavior of the sys-

tem, if the state is unsteady and the volume is not

constant.

P l u g f l o w i n a v e s s e l w i t h c o n s t a n t l en g th .

In this

case, the system section and volume change, although

the length remains constant. It is assumed that u, the

velocity at which the fluid is flowing at a particular

moment, is constant along the vessel and its value is

Qo/S,

which can however change with time. In this

case, it can be easily deduced that the residence time is

a pulse func tion at the mea n value if* = 1.

C o n t i n u o u s s t i r r e d t a n k r e a c t o r .

Making a mass

balance for the tracer, when a pulse injection is used in

a stirred tank, the same re lation indica ted by eq. (23) is

obtained. In this case however 0* and E~ appear,

instead of 0 and

Eo:

E* = exp (- 0" ). (26)

D i s p e r s e d p lu g f l o w w i t h c o n s t a n t l e n g t h .

Assuming

that at any particu lar moment the velocity (u) and the

section (S) of the system are constant along the reac-

tor, where

Q o = Su ,

(although these values can change

with time), and that the dispersion coefficient is con-

sidered constant, the solution obtained would be the

same as in the case of steady state, but using 0* and

E~ instead of 0 and

Eo.

Note that the

( D V / Q o L 2 )

is

the reciprocal of the some times incorrectly called

Peclet number (Levenspiel, 1979).

A P P L I C A T I O N TO T H E S T U D Y O F A S E W A G E S YS T E M

C h a r a c t e r is t ic s o f t h e s y s t e m

The previous equations were used to study the

effect of polluting agents discharged by the Alicante

University Science Faculty (Phase I) on the municipal

sewage system. Four departments (Organic Chem-

istry, Inorganic Chemistry, Physical Chemistry and

Chemical Engineering) as well as some laboratories

from the Biology Section discharge their wastewaters

into the system studied. A tracer injection technique

was used to study the behavior of the system.

NaCl was selected as the tracer, using a solution

near satu ration point and analyzing the level of

sodium ion in the outlet po int of the system. Aroun d

1000 g of sodium ion was used as an aqueous sol ution

which was rapidly discharged into a sink in the stu-

dents' laboratory. At the outlet point, samples were

taken every 5 min and analyzed by flame spectro-

photometry. Results of the tracer concentration

measurements, after subtra cting the Na + content in

the sewage water prior to the run, are presented in

Table 1 for different times. It is possible that other

uncont rolled additions of Na + took place as a conse-

quence of the research and teaching activities carried

out in the laboratories. Table 1 also shows the total

amount of Na introduced. It can be observed that

these values are similar to those calculated by the

integra tion of eq. (1) from the outlet concent ration of

Na +

The tracer stream was injected from the students'

laboratory of the Chemical Engineering Department

because it was the closest point to the sewage system

C a s c a d e o f N e q u a l i d e a l l y s t ir r e d t a n k s w i t h v a r i-

a b l e f l o w a n d v o l u m e , w i t h t h e s a m e r e s i d e n c e t i m e f o r

a l l t h e t a n k s a t a n y t i m e a n d w i t h t o t a l v o l u m e e q u a l t o

N t i m e s t h e l a s t ta n k v o l u m e ( n u m b e r o f t a n k s i s c o n -

s t an t ) .

In this case, it is considered tha t there is a vari-

able flow in the N tanks, a variable total volume and

a variable volume in each one of the tanks, but the

ratio between the volume and the outlet flow (resi-

dence time) is the same for all the tanks. The total

volume, which equals V, is also considered to be equal

to N times the volume of the last tank (VN). The

number of tanks is constant and does not vary with

time.

An expression similar to eq. (24), with 0* and

E~ instead of 0 and

Eo,

can be deduced:

E ~ = I N ( N O * ) N - 1 / ( N -

1) ] exp (-N O*) . (27)

It must be noted that, to calculate eq. (27), the

previous assumptions have been considered.

It can be concluded from the previous deductions

that the general dimensionless relations presented for

steady state and applied to plug flow, ideal stirred

tank, cascade of N-tanks and plug flow with axial

dispersion, can also be applied to systems with con-

stant volume at any situation and to systems with

changing volume only if other conditions occur in the

system.

Table 1. Variation of tracer con-

centration in the different runs

Concentration

Time (s) (kg/m3)

R u n

1 Na amount introduced:

0.98 kg

Na amount calculated by eq. (1):

1.066 kg

Mean residence time calculated by

eq. (6): 1836 s

0 0.0

600 0.0

1200 0.0053

1500 0.0048

1800 0.0117

2100 1.5189

2400 2.5451

2700 1.2749

3000 0.5782

3300 0.357

3600 0.229

3900 0.1639

4200 0.128

4500 0.0953

4800 0.0829

5100 0.0682

5400 0.0655

5700 0.0615

6000 0.0757

6300 0.0625


5/8

R e s i de nc e t i me d i s t r i bu t i on

Table 1 .

(Contd.)

C o n c e n t r a t i o n

Tim e (s ) (kg/m 3)

Run 2 N a + a m o u n t i n t r o d u c e d :

1 .040 kg

N a a m oun t c a l c u l a t e d by e q . (1 ) :

1 .162 kg

Me a n r e s i de nc e t i me c a l c u l a t e d by

eq. (6): 2484 s

0 0 .0

600 0.0

900 0 .0

1200 0.0

1500 0.0

1800 1.4856

2100 1.4476

2400 0.6206

2700 0.3376

3000 0.1786

3300 0.1186

3600 0.0646

3900 0.0446

4200 0 .0356

4500 0.0266

4800 0 .0186

5100 0.0156

5400 0.0786

5700 0.1256

6000 0.0486

6300 0.0225

6600 0.0194

6900 0.0094

7200 0.0044

Run

3 N a a m o u n t i n t ro d u c e d :

0.966 kg

N a a m oun t c a l c u l a t e d by e q . ( 1) :

0.902 kg

Me a n r e s i de nc e t i me c a l c u l a t e d by

eq. (6 ): 2772 s

0 0.0

600 0.0

1200 0.0

1500 1.1512

1800 0.4422

2100 0.0862

2400 0.0517

2700 0 .048

3000 0.0261

3300 0.0057

3600 0.005

3900 0.0036

4200 0 .0

4500 0 .0

4800 0 .0124

5100 0 .0291

5400 0.0138

5700 0.0086

6000 0 .0037

6300 0 .0013

6600 0 .0037

6900 0.0

7200 0.0

a n d w a s t h e r e f o r e t h e p o i n t w h e r e t h e p o l l u t i n g e f fe c t

i s m o s t n o t i c e a b l e ( l es s d i l u t i o n o f t h e t r a c e r s t r e a m ) .

S a m p l e s w e r e t a k e n a t t h e p o i n t w h e r e t h e s y s t e m

s t u d i e d d i s c h a r g e s i n t o t h e g e n e r a l U n i v e r s i t y s e w a g e

f o r uns t e a d y - s t a t e s y s t e ms 227

s y s t e m . T h i s p o i n t w a s s e l e c t e d b e c a u s e i t w a s t h e

o n l y o n e w h e r e i t w a s p o s s i b l e t o m e a s u r e t h e f l o w .

Flow

T h e f l o w o f t h e s y s t e m w a s m e a s u r e d w i t h d i f fi c ul ty .

T h e u s u a l t e c h n iq u e s o f f lo w m e a s u r e m e n t c o u l d n o t

b e u s e d d u e t o t h e d i f f i c u l ty o f a c c e s s t o t h e s e w a g e

s y s t e m a n d t o i t s s h a l l o w n e s s . A v e s s e l w i t h a c a p a c i t y

o f 2 6 1 w a s t h e r e f o r e u s e d t o m e a s u r e t h e f l ow . T h e

o u t l e t p o i n t w a s s e l e c t e d i n o r d e r t o o b t a i n e a s y

a c c e ss a n d g o o d v i si b il it y . F l o w m e a s u r e m e n t s w e r e

t a k e n e v e r y 5 m i n f o r a p e r i o d o f 1 0 0 - 1 2 0 m i n . T h e

r e s u l t s a r e p r e s e n t e d i n F i g . 2 .

Numerical treatment of the results

V a l u e s o f t h e o u t l e t t r a c e r c o n c e n t r a t i o n s a r e

p l o t t e d v s t i m e i n F i g . 3 . I f t h e s e r e s u l t s a r e u s e d t o

o b t a i n c u r v e E t o e s t a b l i s h t h e r e s i d e n c e t im e d i s t r i -

b u t i o n , d i f f e r e n t c u r v e s o b t a i n e d a t d i s t i n c t d a y s c a n

b e s h o w n i n F i g . 4 . T h e r e a s o n f o r th i s b e h a v i o r i s t h e

out le t f low x (1E +3) (m3/s )

2

1 . 5

0 . 5

/ ,,, ~ . ~ - . ~ ~ , ~

\k ' / & :

J

i

4 6

t ime x 1E-3 ) s )

I ~ r u n l ~ , - r u n 2 ~ r u n 3 J

Fig. 2 . Out le t f low vs t ime.

c o n c e n t ra t i o n ( k g /m 3 )

3

2.5

2

1.5

1

0.5

0

2 4 6

t ime

x 1E-3) s)

I ~ n ~ n l * ~ n 2 ~ r o n _ ~

Fi g. 3 , O u t l e t c on c e n t r a t i on v s t i me .


6/8

228

E x ( 1 E + 3 ) ( l / s )

2 .5

1.5

0.5

0 2 ~ 4 - 6 8

t im e x (1 E-3) (s)

I ~ - r u n l ~run2 ~ r u n 3 1

J . F E R N A N D E Z - S E M P E R E et al.

E./Oo I m3)

2.5

~ i ,

I ,

' l

1.5 ~

i

0 . 5

0 ~ . . . .

0

Fig. 4. Residence time function (E) vs time.

k ,

2 4 6 8 10

f Q o d , , m 3 ,

j O r u n l r u n 2 ~ r u n 3 ]

t

Fig. 5. Residence time function (E/Qo) vs ~oQodt .

cont inuous change in the system flow (as can be seen

in Fig. 2) as well as in the system volume.

In order to predict the behavior of the system, the

above-described treatment was used. The tracer

amoun t (M) and the average residence time (/-) were

calcula ted for equal time intervals using eqs (1) and (6)

(Table 1). Next, a new E function was obtained from

eq. (5).

Initially it was assumed that the volume in the

system was constant, although the flow is variable.

However, when E / Q o was plotted vs So Q o d t , (Fig. 5)

it was found that the curves for different experiments

did not coincide. This could be due to a variable

system volume. On considering the average residence

time and the range of outlet flows, it can be deduced

that the system volume is different in each run. Conse-

quently, the volume was then considered as a function

of the system outlet flow:

V = a Qo . (28)

An explanation of this variation is presented in Ap-

pendix A.

According to eq. (26), values of E V / Q o were plotted

vs the dimensionless time ( 0 * = So ( Q o / V ) d t using the

value of V given by eq. (28). Parameters a and b were

optimized by means of a modified simplex method in

order to obtain an average dimensionless time if*

close to unity for each run (Fig. 6). The expression

obtained for the system volume was

V = 286.31(Qo)O.7154 (29)

where V is expressed in m 3 and

Qo

in m3/s. The

exponent 0.7154 is similar to that deduced in Appen-

dix A.

In Fig. 6, it can be observed that there is a great

similarity in the value of 0* in the three experiments,

as well as in the shape of the three curves. Most of the

tracer appears in a big peak (at values of 0* between

0.7 and 0.9) and later, a much smaller peak appears.

Note that there is logically a coincidence of the

greatest peak around 0* = 1 but, at the same time,

E*=EV/Qo

5

0.5 1 1.5 2 - 2.5 3 3.5

] O r u n l r u n 2 ~ r u n 3 1

0 *

Fig. 6. Residence time function (E*) vs 0".

there is a coincidence, in runs 2 and 3, of the smallest

peak. In r un 1, with a smaller value of rate flow, tak ing

of samples was stopped before the appear ance of the

second maximum. However, for the last experimental

points in run 1, it can be observed that there is an

increase of the E* values, according to the second

maximum that appears in the other runs. The coincid-

ence of the second peak has not been introduced in

the model and therefore corroborates the utility of the

proposed model.

The maximum value of E v / Q o in Fig. 6 allows the

maximum concentration that would be obtained in

the system to be calculated if an amount M of the

polluting agent were discharged into it. Thus, taking

into account eq. (5):

E = C Q o / M

and therefore

E V /Q o = C V / M . (30)

So, the maximum concentration can be obtained

from eq. (31):

Cm =x = ( M / V ) ( E V /Q o ) . . . (31)


7/8

Residence t im e d is t r ibut ion for u ns teady-s ta te sys tems

Ta ble 2. Characteristics of the system stu died

229

R u n

Flow range M ax im um t racer M ax im um t race r

( x 1 0 4 ) concen t ra t ion concen t ra t ion / tra ce r

(m3/s) (kg/m 3) am oun t ( l /m 3)

Dispers ion

Dispersion coefficient

module (m2/s )

1 2.6 -6 2.54 2.4

2 6-1 4.3 1.49 1.3

3 11.6-16.6 1.15 1.3

0.038 0.553

0.021 0.351

0.012 0.245

A t t h e s a m e t im e , f r o m F i g . 6 , t h e v a l u e o f t h e E s

d i s p e rs i o n m o d u l e

[D/uL = D V/(Qo

L 2 ) ] a s a f u n c -

t i o n o f t h e s y s te m f lo w ( T a b l e 2 ) c a n b e c a l c u l a t e d . E *

T h e g r e a t e r t h e f l o w , t h e s m a l l e r t h e d i s p e r s i o n

m o d u l e . T h e v a l u e o f L i s a r o u n d 1 9 5 m . C o n s i d e r i n g L

t h e m e a n v a l u e s f o r t h e f lo w r a t e r a n g e in d i c a t e d i n M

T a b l e 2 a n d t h e c o r r e s p o n d i n g v a l u e s o f V , t h e d i s p e r - N

s i o n c o e f f i c ie n t s w e r e e s t i m a t e d a n d a r e p r e s e n t e d i n Q i

T a b l e 2 . I t c a n b e o b s e r v e d t h a t t h e d i s p e r s i o n c o e f f i -

Qo

c i e n t d e c r e a s e s w h e n t h e m e a n f lo w is g r e a t e r . A I - S

t h o u g h t h e a i m o f t h i s p a p e r i s n o t t o o b t a i n th e t

r e l a t i o n s h i p b e t w e e n t h e d i s p e r s i o n c o e ff i ci e n t a n d t h e i

f lo w i n s e w a g e s y s te m s , a n e x p l a n a t i o n b a s e d o n s o m e u

d a t a f o u n d in t h e l i t e r a t u r e (L e v e n s p i e l , 1 96 2, 19 79 ) i s V

p r e s e n t e d in A p p e n d i x B . T a b l e 2 a l so s h o w s th e x

m a x i m u m t r ac e r c o n c e n t r a t i o n o b t a i n e d a n d t h e r a t io z

( m a x i m u m c o n c e n t r a t i o n / t r a c e r a m o u n t ) , w h e r e t h e

v a l u e s o f M c o n s i d e r e d a r e t h o s e o b t a i n e d f r o m i n t e -

g r a t i o n o f e q . ( 1 ).

A l t h o u g h t h e p r o p o s e d m o d e l d o e s n o t e x a c t l y

o b t a i n t h e s a m e c u r v e f o r d i f f e r e n t c o n d i t i o n s , i t i s

c a p a b l e o f p r e d i ct i n g a p p r o x i m a t e l y p e a k s o r m a x i m a

o f c o n c e n t r a t i o n , w h o s e d i s p e r s i o n i s a f u n c t i o n o f t h e

o p e r a t i n g f lo w .

CONCLUSIONS

F o r c l o s e d - c l o s e d s y s te m s , w h e r e t h e i n l e t f lo w , t h e

o u t l e t f lo w a n d t h e v o l u m e o f a n i n c o m p r e s s i b l e f l u i d

v a r y w i t h t i m e , i t i s u s e f u l t o d e f i n e a r e s i d e n c e t i m e

f u n c t i o n E a n d a d i m e n s i o n l e s s r e s id e n c e t im e f u n c -

t i o n E * . T h e r e s i d e n c e t i m e f u n c t i o n E h a s a m e a n i n g

s i m i l a r to t h a t o f s te a d y - s t a t e s y st e m s . T h i s m e a n s

t h a t , f o r a v o l u m e o f f l u i d e n t e r i n g t h e s y s t e m a t t = 0 ,

E d t i s t h e f r a c t i o n o f f l u i d v o l u m e w h i c h l e a v e s t h e

s y s t e m b e t w e e n t a n d t + d t . T h e d i m e n s i o n l e s s re s i -

d e n c e t i m e f u n c t i o n E * i s d e f i n e d a s

E V/Qo

a n d

p l o t t e d v s th e d i m e n s i o n l e s s t i m e 0 " , w h i c h e q u a l s

S~(Qo/V)dt.

U s i n g t h e p r e v i o u s d e f i n i ti o n s , d a t a o b t a i n e d i n

a s e w a g e s y s te m a n d c o r r e s p o n d i n g t o t h r e e d i ff e r en t

c u r v e s

E =f(t)

w i t h d i f f e r e n t f l o w r a n g e s , c a n b e

t r a n s f o r m e d i n t o t h r e e v e r y s i m i l ar c u r v es

E* =f(O*)

w i t h t h e c o i n c i d e n c e o f t h e m a x i m a .

C

C o

D

E

NOTATION

t r a c e r c o n c e n t r a t i o n , k g / m 3

i n i t ia l t r a ce r c o n c e n t r a t i o n , k g / m 3

d i s p e r s i o n c o e f f i c i e n t , m 2 / s

r e s i d en c e t i m e d i s t r i b u t i o n f u n c t i o n , s -

d i m e n s i o n l e s s r e s id e n c e t i m e d i s t r i b u t i o n

f u n c t i o n

d i m e n s i o n l e s s r e s i d e n ce t i m e d i s t r i b u t i o n

f u n c t i o n w h e n t h e s y s te m v o l u m e i s v a r i a b l e

s y s t e m l e n g t h , m

t r a c e r t o t a l a m o u n t , k g

n u m b e r o f s t ir r ed t a n k s

i n l e t f l o w , m 3 / s

o u t l e t f l o w , m 3 / s

s y s t e m s e c t i o n , m E

t ime , s

m e a n r e s i d en c e t i m e , s

f l u i d v e l o c i t y , m / s

s y s t em v o l u m e , m 3

d i s t a n c e , m

d i m e n s i o n l e s s l e n g t h

Greek letters

0 d i m e n s i o n l e s s t i m e [ e q . ( 2 1 ) ]

0 * d i m e n s i o n l e s s t i m e [ e q . ( 2 6 ) ]

R E F E R E N C E S

Calu, M. P. and Lameloise, M . L. , 1986, In terpre ta t ion de

mesures de dispers ion des temps de se jour da m des 6coule-

ments de masse volumique variable . Appl ica t ion f i la

mode l i s a t ion d ' evapora teu rs

f i f l o t

montant de sucrer ie .

Entropie

22 (128) 13-22.

Den bigh, K. G. and T urner , J . C . R. , 1984, Chemical Reactor

Theory.

An Introduction,

Cambridge Univers i ty Press ,

Cambridge.

Dickens, A. W., Mackley, M. R. and Williams, H. R., 1989,

Experimenta l res idence t ime dis t r ibut ion measurements

for unsteady flow in baffled tubes.

Chem. Engno Sci. 44,

1471-1479.

Fan , L. T., Fan, L. S. and Nass ar, R . F., 1979, A stochastic

model of the uns teady s ta te age dis t r ibut ion in a f low

system.

Chem. Engno Sci. 34,

1172-1174.

Fogler, H. S., 1992,

Elements of Chemical Reaction Engineer-

ing.

Prent ice-Hal l , E nglewood Cliffs, NJ .

From ent , G. F. an d Bischoff , K. B., 1979, Chemical Reactor

Analysis and Design. Wiley, New York.

Levenspiel, O., 1962,

Chemical Reaction Engineering.

Wiley,

New York.

Levenspiel, O., 1979, The

Chemical Reactor Omnibook.

O S U

Book Stores, Corvallis , OR.

Na um an, E. B., 1969, Res idence t ime dis t r ib ut ion theory for

unsteady stirred tank reactors. Chem. Engng Sci. 24,

1461-1470.

Schwartz, S. E., 1979, Residence time in reservoirs und er

non-s teady-s ta te condi t ions : appl ica t ion to a tmospheric

SO2 and aerosol sulfate.

Tellus

31, 530-547.

Smith, J . M., 1981,

Chemical Engineering Kinetics.

M c G r a w -

Hill, New York.

Vaccari, D. A., Fagedes, T. and Lo ngtin , J ., 1985, Ca lcula tion

of me an ce ll res idence t ime for un s teady-s ta te ac t iva ted

sludge systems.

Biotechnol. Bioengng

27, 695-703.


8/8

23O

We s t e r t e r p , K . R . , Va n Swa a i j , W. P . M. a nd Be e na c ke r s ,

A. A. C. M ., 1993, Chemical React or Design and Operation.

Wiley, New York.

A P P E N D I X A

Cons i de r t he s y s t e m be t we e n t he d i s c ha r ge po i n t o f t he

s a t u r a t e d s od i um s a l t a nd t he e x i t po i n t o f t he s e wa ge

s ys t e m, whe r e i t wa s pos s i b l e t o me a s u r e t he f l ow a n d t a ke

s a mpl e s f o r a na l y s i s . The me c ha n i c a l e ne r gy l o s s AF ( J / kg )

a pp r ox i ma t e l y e qua l s g Az , whe r e g i s t he g r a v i t y a c c e l e r -

a t i on ( m2 / s ) a nd Az t he d i f f e re nc e be twe e n t he he i gh t s o f t he

t wo po i n t s c ons i de r e d ( m) . Cons e que n t l y f o r d i f f e r e n t f l ow

r a t e s , t he me c ha n i c a l e ne r gy l o s s i s c ons t a n t . As s um i ng t ha t

t he c i r c u l a t i on r e g i me i s t u r bu l e n t , i t c a n be wr i t t e n t ha t

2f u 2 L

A F = - - (A1)

D~

wh e r e f i s t he f r a c t i on f a c t o r , u i s t he m e a n ve l oc i ty ( m/ s ), L i s

t he t o t a l l e ng t h be t we e n t he t wo po i n t s c ons i de r e d ( m) a nd

D h i s t he hyd r a u l i c d i a m e t e r ( m) . The s e wa ge s y s t e m i s

f o r me d by l a r ge c i r c u l a r c onc r e t e t ube s , i n wh i c h w a s t e wa t e r

c i r c u l a te s oc c upy i ng on l y t he bo t t om o f t he cy l i nde r . F i gu r e

F i g . 7. D i a g r a m o f the s e c t i on oc c up i e d by t he wa s t e wa t e r .

J. FERNANDEZ-SEMPERE et al.

7 s hows a d i a g r a m o f t he s e c t i on . As s umi ng t ha t t he a ng l e

i s no t ve r y l a r ge , i t c a n be de duc e d t ha t ( whe r e c t i s i n

r a d i a ns a nd h - ~ b i s a s s ume d)

s in ~ ~ ct ~ h/R ~- b/R (A2)

b 2 = a ( 2 R - a ) - a 2 R ( A 3)

p e r i m e t e r = p = 2R (A4)

sect ion = S = Rc t a (A5)

hyd r a u l i c d i a me t e r D ~ = 4 S/p = 2a (A6)

F r o m e q s ( A 2 )- ( A 6 ), o n e o b t a i n s

/ S 2 ~ 1 /3

D h = 2 \ ~ - ~ j . (A 7)

I n t r oduc i ng e q . ( A7) i n e q . ( A1) a nd a s s umi ng t ha t t he

vo l um e o f t he s y s t e m c a n be c ons i de r e d V = SL, the follow-

i ng e xp r e s s i on c a n be wr i t t e n :

V = (L11 2R )l/s QO.TS. (A8)

A s s u m i n g t h a t t h e f a c t o r f d o e s n o t c h a n g e c o n s i d e r a b l y

wi t h t u r bu l e nc e , t h i s t he o r e t i c a l r e l a t i on a g r e e s w i t h t he

e x p e r i m e n t a l p o t e n t i a l r e l a ti o n b e t w e e n t h e s y s t e m v o l u m e

V a nd t he ou t l e t vo l um e t r i c fl ow , w i t h a n e xpo ne n t e qua l t o

0 .7154, which i s very s imi lar to the theore t ica l va lue 0 .75

from eq. (A8).

APPEN D I X B

O n c on s i de r i ng t he ( de fi ne d ) hyd r a u l i c d i a m e t e r a nd o t h e r

r e l a t i ons c ons i de r e d i n e qs ( A2) - ( A7) , i t is de duc e d t ha t t he

R e yn o l ds num be r i s a r oun d 6 000, 11 ,000 a nd 15 ,000 f o r r uns

1, 2 and 3 , respectively . This m ean s tha t the regime i s prob-

a b l y t u r b u l e n t i n t h e t r a n s i t i o n z o n e , f r o m l a m i n a r t o t u r b u -

l e n t . Le ve ns p i e l ( 1962 ) p r e s e n t e d a va r i a t i on o f d i s pe r s i on

coeff ic ients for f lu ids in p ipes , vs Reynolds number . From

t h i s r e la t i on , i t c a n be de duc e d t ha t t he d i s pe r s i on c oe f f ic i e n t

c a n d e c r e a s e w h e n t h e R e y n o l d s n u m b e r i n c r e a s e s i n t h e

t u r bu l e n t r e g i me i n t he t r a ns i t i on z one . The va l ue s o f t he

d i s pe r s i on c oe f fi c i en t s a r e h i gh , p r oba b l y a s a c ons e que nc e o f

s t one s , me t a l s , o r o t he r obs t a c l e s i n s i de t he c onc r e t e t ube

wh i c h c a us e a dd i t i on a l m i x i ng o f t he e l e me n t s o f f lu id .

krd important

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