Nonequilibrium Systems in Natural Water Chemistry

7 Use of Inert Tracer Data to Estimate the Course of Reaction in Geometrically Complex Natural Flows

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L L O Y D A. S P I E L M A N and FRANCOIS BRIERE Division of Engineering and A p p l i e d Physics, H a r v a r d University, Cambridge, Mass. 02138

The dispersion

model, while

suitable

data from geometrically

simple

plex

the

geometries.

originally

developed

application and occur

Here by

to steady

extended

to

in modeling

time

Danckwerts transient

and

control.

to

Zwietering is

loadings

which

Sample

computer

are based on tracer responses

of two flows:

theoretical,

the

Results

a natural

stream.

sented for both slug and step inputs of reactants ing according With

second

is bracketed minimum

to both first and second order by upper

and complete

the bounds

reaction,

the

and lower segregation.

merge to give a single

for

described

calculations

other

com-

interpretation,

reactors,

reactant

for pollution

tracer

is not suited

residence

state industrial

treat

for interpreting

flows,

are

one pre-

disappear-

order rate

expressions.

downstream

response

bounds

corresponding

For first order response

to

reaction,

curve.

T n m o d e l i n g r i v e r s , streams, a n d other n a t u r a l flows for p o l l u t i o n c o n t r o l , A

i t is often necessary to estimate d o w n s t r e a m c o n c e n t r a t i o n profiles, i n

d i s t a n c e a n d t i m e , o f substances w h i c h s i m u l t a n e o u s l y u n d e r g o r e a c t i o n a n d c o n v e c t i o n after t h e i r i n t r o d u c t i o n u p s t r e a m . T h e s e substances m i g h t b e d i s s o l v e d or s u s p e n d e d p o l l u t a n t s , m i c r o o r g a n i s m s , o x y g e n , o r other r e l e v a n t constituents o f the aqueous e n v i r o n m e n t . T o p r e d i c t c o n c e n t r a t i o n profiles, i t is d e s i r e d to c o m b i n e i n d e p e n d ent l a b o r a t o r y r e a c t i o n rate d a t a w i t h e m p i r i c a l d a t a o n the h y d r o d y n a m i c s o f the flow. T h e p r o b l e m o f i n t e r p r e t i n g l a b o r a t o r y r e a c t i o n d a t a to o b t a i n rate expressions w h i c h are v a l i d w i t h i n the c o m p l e x n a t u r a l s y s t e m is i n itself a difficult task b e y o n d the scope o f this w o r k ; i t is here 194 Hem; Nonequilibrium Systems in Natural Water Chemistry Advances in Chemistry; American Chemical Society: Washington, DC, 1971.

7.

spiELMAN A N D BRiERE

Use of Inert Tracer

195

Data

a s s u m e d that v a l i d c o n c e n t r a t i o n - d e p e n d e n t expressions for rates of r e a c t i o n are a v a i l a b l e . W i t h the g i v e n rate expression, d e t a i l e d t h e o r e t i c a l or e m p i r i c a l k n o w l e d g e of the s p a t i a l v e l o c i t y d i s t r i b u t i o n w o u l d p r o m o t e use of t h e equations of c o n v e c t i v e t r a n s p o r t ( I )

to e v a l u a t e c o n c e n t r a t i o n profiles

a n d give a comprehensive treatment, but direct in-stream measurements r e q u i r e d to o b t a i n s u c h v e l o c i t y d a t a are p r o h i b i t i v e l y costly a n d n o t g e n e r a l l y a v a i l a b l e . I n e r t tracer studies for t h e f l o w are far m o r e p r a c t i c a l a n d economical but do not convey complete h y d r o d y n a m i c information. T r a c e r studies are n o r m a l l y c a r r i e d out b y i n t r o d u c i n g a c o n v e n i e n t l y Downloaded by SUFFOLK UNIV on January 18, 2018 | http://pubs.acs.org Publication Date: June 1, 1971 | doi: 10.1021/ba-1971-0106.ch007

m o n i t o r e d solute ( s u c h as a r a d i o a c t i v e i s o t o p e ) , u s u a l l y as a s l u g , a n d m e a s u r i n g its c o n c e n t r a t i o n response w i t h t i m e at stations d o w n s t r e a m (2).

F o r reaches of the flow w h i c h are n o t g e o m e t r i c a l l y c o m p l e x , t h e

longitudinal dispersion model

c a n often

be

applied, using

measured

t r a c e r response curves to extract n u m e r i c a l d i s p e r s i o n coefficients.

These

c a n t h e n b e i n s e r t e d b a c k i n t o the d i s p e r s i o n e q u a t i o n w i t h a r e a c t i o n rate t e r m to estimate the concentrations of r e a c t i n g species w h i c h are i n t r o d u c e d u p s t r e a m i n a n assigned f a s h i o n ( 3 , 4). flow

F o r regions of the

w h i c h are g e o m e t r i c a l l y c o m p l e x , h o w e v e r , tracer response

curves

c a n differ i n shape g r e a t l y f r o m those e x p e c t e d f r o m the d i s p e r s i o n m o d e l a n d a n alternate m e t h o d of i n t e r p r e t i n g t h e m m u s t b e sought.

Illustra-

tions of s u c h p o s s i b l e c o m p l e x i t i e s are s h o w n i n F i g u r e 1. I n t e r p r e t i n g t r a c e r response curves as residence t i m e d i s t r i b u t i o n s , a l o n g w i t h considerations of c o m p l e t e a n d m i n i m u m segregation i n the flow,

p r o v i d e s a r i g o r o u s basis f o r t r e a t i n g g e o m e t r i c a l l y c o m p l e x

flows.

This approach was pioneered b y Danckwerts (5) and Zwietering (6)

for

a p p l i c a t i o n to n o n i d e a l i n d u s t r i a l reactors. I n d u s t r i a l reactors over w h i c h the engineer

has c o n s i d e r a b l e

geometrical

control behave

more

pre-

d i c t a b l y t h a n most n a t u r a l flows a n d often c a n b e t r e a t e d t h e o r e t i c a l l y ; thus, a p p l i c a t i o n of the s e m i e m p i r i c a l residence t i m e a p p r o a c h to n a t u r a l systems c o u l d b e e v e n m o r e significant t h a n its o r i g i n a l l y i n t e n d e d use. W h i l e r e s i d e n c e t i m e t h e o r y s h o u l d b e a p p l i c a b l e to a w i d e v a r i e t y of g e o m e t r i c a l l y c o m p l e x flows, i t possesses i n h e r e n t t h e o r e t i c a l l i m i t a t i o n s w h i c h s h o u l d be discussed. A n i m p o r t a n t one is that i n p u t a n d m o n i t o r i n g stations s h o u l d c o r r e s p o n d to constrictions of the

flow

(Figure

1)

w h e r e concentrations across the s t r e a m w i d t h m a y b e c o n s i d e r e d as u n i f o r m ( a l t h o u g h m u c h c a n also be s a i d a b o u t t h e d i s t r i b u t i o n of

concen-

trations b e t w e e n s t a t i o n s ) . T h i s is seen to b e n o great h a n d i c a p , h o w e v e r , w h e n c o m p a r e d w i t h the o n e - d i m e n s i o n a l d i s p e r s i o n m o d e l w h i c h assumes a u n i f o r m c o n c e n t r a t i o n across the s t r e a m w i d t h f o r a l l p o i n t s a l o n g the flow. A n o t h e r l i m i t a t i o n occurs i f the r e a c t i o n d e p e n d e n c e o n c o n c e n t r a t i o n is g r e a t l y different f r o m first o r d e r ; t h e n , w i t h

knowledge

of the residence t i m e d i s t r i b u t i o n , one c a n o n l y b r a c k e t the c o n c e n t r a t i o n

Hem; Nonequilibrium Systems in Natural Water Chemistry Advances in Chemistry; American Chemical Society: Washington, DC, 1971.

196

NONEQUILIBRIUM

SYSTEMS

IN

N A T U R A L

WATERS


MONITOR

Figure 1. Possible natural flows having complex geometry and tracer response. Input and monitoring stations are at constrictions. Upstream of station A , the dispersion model might apply. w i t h least u p p e r a n d greatest l o w e r b o u n d s . O n l y for the i m p o r t a n t s p e c i a l case of first o r d e r

( l i n e a r ) r e a c t i o n is the d o w n s t r e a m

concentration

u n a m b i g u o u s l y d e t e r m i n e d for a specified i n e r t t r a c e r response. t i g h t l y c o n c e n t r a t i o n is b r a c k e t e d f o r a g i v e n system d e p e n d s

How o n the

p a r t i c u l a r f o r m of the residence t i m e ( t r a c e r r e s p o n s e ) c u r v e , t h e d e g r e e of n o n l i n e a r i t y of the r e a c t i o n , a n d the extent of r e a c t i o n o c c u r r i n g b e t w e e n stations. T h a t the tracer response c u r v e does n o t i n g e n e r a l fix system b e h a v i o r u n a m b i g u o u s l y for n o n l i n e a r reactions is r e a d i l y d e m o n s t r a t e d b y

con-

s i d e r i n g the p a r t i c u l a r system c o n s i s t i n g of a n i d e a l p l u g flow vessel i n alternate s e q u e n c e w i t h a n i d e a l l y - s t i r r e d vessel. A s i l l u s t r a t e d i n F i g u r e 2, i n t e r c h a n g i n g the o r d e r of the t w o vessels leaves the s h i f t e d e x p o n e n t i a l response to a s l u g i n p u t of t r a c e r u n c h a n g e d ; y e t f o r reactions w i t h o r d e r greater t h a n u n i t y , s t r a i g h t f o r w a r d c a l c u l a t i o n s c o n f i r m that the c o n f i g u r a t i o n h a v i n g the p l u g flow vessel first y i e l d s m o r e extensive r e a c t i o n . T h e s i t u a t i o n is r e v e r s e d for r e a c t i o n orders less t h a n u n i t y . B o t h a r r a n g e ments g i v e the same response i f the r e a c t i o n is first o r d e r . B e c a u s e of the a m b i g u i t y of b e h a v i o r w h i c h r e m a i n s w i t h k n o w l e d g e of a s y s t e m s tracer response, i t c a n b e c o n c l u d e d t h a t close a g r e e m e n t of i n e r t tracer response curves to the f o r m e x p e c t e d

f r o m the d i s p e r s i o n


7.

spiELMAN A N D BRiERE

Use of Inert Tracer

197

Data

m o d e l does not c o n c l u s i v e l y c o n f i r m the v a l i d i t y of t h a t m o d e l

between

t w o m o n i t o r i n g stations; thus, a p p l i c a t i o n of the d i s p e r s i o n m o d e l to n o n l i n e a r reactions s h o u l d b e m a d e w i t h c a u t i o n e v e n i f i n e r t t r a c e r response appears i d e a l . It is s u b s e q u e n t l y s h o w n h o w a n y g i v e n i n e r t tracer r e sponse o f a system s t i l l p e r m i t s a r a n g e of p o s s i b l e responses f o r n o n l i n e a r r e a c t i o n a n d h o w the m i n i m a l b o u n d s o n t h a t r a n g e c a n b e Elements of Residence Time

computed.

Theory

S i n c e the f o u n d a t i o n s of r e s i d e n c e t i m e t h e o r y are r i g o r o u s l y g i v e n Downloaded by SUFFOLK UNIV on January 18, 2018 | http://pubs.acs.org Publication Date: June 1, 1971 | doi: 10.1021/ba-1971-0106.ch007

elsewhere ( 5 , 6, 7 ) , o n l y those features w h i c h are essential t o the present t r e a t m e n t w i l l b e g i v e n here. T h e residence t i m e d i s t r i b u t i o n ( r e s i d e n c e t i m e f r e q u e n c y f u n c t i o n ; exit age d i s t r i b u t i o n ) , / ( * ) , is d e f i n e d s u c h that f(t)dt

is the f r a c t i o n of f l u i d at a n y instant l e a v i n g the system, h a v i n g

spent t i m e b e t w e e n

t and t +

dt w i t h i n the system.

The cumulative

residence t i m e d i s t r i b u t i o n is

o

o

CO

f(t)

Figure 2. Ideal plug flow and perfectly mixed vessels in alternate sequence. Both arrangements exhibit the same response for inert tracers and first order reactions but not for nonlinear reactions.


198

NONEQUILIBRIUM SYSTEMS IN N A T U R A L WATERS

7 f(t')dt'

F(t)

(1)

corresponds to the f r a c t i o n of fluid at a n y instant l e a v i n g t h e s y s t e m ,

h a v i n g spent t i m e shorter t h a n t w i t h i n the system. A r e l a t i o n of f u r t h e r i m p o r t a n c e is


(2)

T h a t is, t h e first m o m e n t of the residence t i m e f r e q u e n c y f u n c t i o n residence t i m e )

(mean

equals t h e q u o t i e n t of the system v o l u m e V a n d the

constant v o l u m e flow rate Q t h r o u g h the system. T h e f u n c t i o n f(t)

is s t r a i g h t f o r w a r d l y r e l a t e d to t h e e x p e r i m e n t a l

t r a c e r response c u r v e ; for a s l u g i n p u t , t h e d o w n s t r e a m c o n c e n t r a t i o n t i m e c u r v e is p r o p o r t i o n a l to the f u n c t i o n f(t).

vs.

R e s p o n s e curves f o r d i f -

ferent m o d e s of i n p u t t h e o r e t i c a l l y c o n v e y e q u i v a l e n t i n f o r m a t i o n , a l t h o u g h c e r t a i n i n p u t s are e x p e r i m e n t a l l y c o n v e n i e n t to c a r r y out.

Thus,

r e d o i n g tracer experiments for different i n p u t s gives n o a d d e d i n f o r m a t i o n a b o u t the

flow.

Extremes of Mixing Residence Time

for

an Arbitrary

Specified

Distribution

Z w i e t e r i n g w a s a b l e to p r o v e that t h e t w o c o n c e p t u a l reactor figurations

con-

s h o w n i n F i g u r e 3, e a c h h a v i n g the same specified a r b i t r a r y

residence t i m e d i s t r i b u t i o n , e x h i b i t o p p o s i t e extremes of m i x i n g a n d thus give bounds on chemical conversion. flow

vessel h a v i n g c o n t i n u o u s

E a c h c o n f i g u r a t i o n is a n i d e a l p l u g

d i s t r i b u t i o n of respective

side exits

entrances w h i c h are g o v e r n e d to g i v e t h e a r b i t r a r y specified time distribution. Zwietering showed

or

residence

that the l o w e r c o n f i g u r a t i o n

in

F i g u r e 3 corresponds to t h e m i n i m u m degree of segregation for a g i v e n residence

t i m e d i s t r i b u t i o n , w h i l e the u p p e r

d i a g r a m corresponds

to

c o m p l e t e segregation or d e g r e e of segregation u n i t y . T h e degree of segreg a t i o n w a s i n t r o d u c e d b y D a n c k w e r t s a n d is a m e a s u r e of the extent to w h i c h m o l e c u l e s e n t e r i n g t h e system at a p a r t i c u l a r t i m e r e m a i n together, n o t m i x i n g w i t h m o l e c u l e s w h i c h enter at o t h e r times. W h i l e this c o n c e p t arises i n the f u n d a m e n t a l s of the theory, its precise d e f i n i t i o n o r n u m e r i c a l e v a l u a t i o n is n o t n e e d e d for a p p l i c a t i o n of the t h e o r y a n d w i l l not

be

g o n e i n t o here. Z w i e t e r i n g a l t e r n a t e l y interprets c o m p l e t e a n d m i n i m u m segregation

for the specified

"earliest" m i x i n g , respectively.

residence

t i m e f u n c t i o n as " l a t e s t " a n d

T h e time coordinate a shown i n F i g u r e 3


7.

SPIELMAN AND B R i E R E

Use of Inert Tracer

199

Data

c o r r e s p o n d s to t h e age ( t i m e i n t e r v a l after e n t e r i n g ) o f a fluid e l e m e n t i n t h e s y s t e m , a n d t h e c o o r d i n a t e λ is t h e l i f e e x p e c t a t i o n ( t i m e i n t e r v a l before l e a v i n g ) o f a n element. COMPLETE SEGREGATION

c (t) 0


lmnnmmm

c (t) e

MINIMUM SEGREGATION C (t) n

1111111111Π1ΤΤΤ

c (t) e

Figure 3. Conceptual flows corresponding to ex tremes of complete and minimum segregation. The distributed entrances and exits are governed to give the same arbitrary specified residence time distribution for each system. Analysis

for Complete

Segregation

I n t h e case o f c o m p l e t e s e g r e g a t i o n , t h e v o l u m e dv o f a n e l e m e n t o f fluid w i t h age b e t w e e n a a n d a + da is

* = V τ

F(oL)]doL

(3a)

T h e r e l a t i o n b e t w e e n t h e v o l u m e c o o r d i n a t e ν ( F i g u r e 3 ) a n d α is t h e n


200

NONEQUILIBRIUM

α

[1 -

SYSTEMS

IN

N A T U R A L

WATERS

(3b)

F{ot)W

i n w h i c h F is g i v e n b y E q u a t i o n 1. T h e equations d e s c r i b i n g r e a c t i o n i n the system w e r e d e r i v e d p r e v i o u s l y o n l y f o r the steady state w i t h a constant i n c o m i n g c o n c e n t r a t i o n . F o r i n d u s t r i a l reactors that case is of u t m o s t i m p o r t a n c e , b u t f o r n a t u r a l systems transient situations are e q u a l l y i m p o r t a n t , a n d the t r e a t m e n t is e x t e n d e d here to i n c l u d e a n a c c u m u l a t i o n t e r m . F o l l o w i n g Z w i e t e r i n g , let c(a,t) d e n o t e the c o n c e n t r a t i o n of r e a c t i n g substance i n the system Downloaded by SUFFOLK UNIV on January 18, 2018 | http://pubs.acs.org Publication Date: June 1, 1971 | doi: 10.1021/ba-1971-0106.ch007

as i t d e p e n d s o n b o t h p o s i t i o n ( t h r o u g h E q u a t i o n 3 b ) let R(c)

a n d t i m e t, a n d

b e the c o n c e n t r a t i o n - d e p e n d e n t rate of d i s a p p e a r a n c e o f reactant

per unit volume.

C o n s i d e r i n g n o w a v o l u m e e l e m e n t of the system a n d

u s i n g E q u a t i o n 3b, the q u a n t i t y of reactant l e a v i n g t h r o u g h side exits is Qc(a,t)f(a)da.

A d e t a i l e d m a t e r i a l b a l a n c e o n the v o l u m e e l e m e n t gives

the p a r t i a l d i f f e r e n t i a l e q u a t i o n for the c o n c e n t r a t i o n profile as dc

, dc

Fa + M

D

-

=

R

ν

/ (

C

(4)

)

T h e t i m e - d e p e n d e n t c o n c e n t r a t i o n i n the c o m b i n e d exit s t r e a m is

(5)

i n w h i c h c ( « , f ) is the s o l u t i o n of E q u a t i o n 4. E q u a t i o n 4 is to b e s o l v e d u n d e r the c o n d i t i o n s : b e f o r e t i m e t = system, a n d after t =

0, n o reactant is present i n the

0, the i n c o m i n g c o n c e n t r a t i o n c

0

is a n a r b i t r a r y

specified f u n c t i o n of t i m e . T h a t is, c(a,i) = 0 for t < 0

(6a)

= 0 for t

0

(6c)

E q u a t i o n 4 is q u a s i l i n e a r a n d c a n b e s o l v e d r e a d i l y b y the m e t h o d of characteristics ( 8 ) .

Its s o l u t i o n u n d e r c o n d i t i o n s 6 a , 6 b , a n d 6c is g i v e n

b y the i m p l i c i t r e l a t i o n


7.

spiELMAN

201

Use of Inert Tracer Data

BRiERE

A N D

F o r a disappearance reaction of order η ^

1, R(c)

= fcc* a n d E q u a t i o n

7 gives c(

Nonequilibrium Systems in Natural Water Chemistry

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