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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
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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)
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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
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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
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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 .
8/12/2019 krd important
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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)
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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.
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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 .