9.nin ideal flow

7/27/2019 9.Nin Ideal Flow

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NON-IDEAL FLOW

Residence Time Distribution

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SCOPE:

Design of non-ideal reactors

Identify the possible deviations

Measurement of RTD

Quality of mixing

Models for mixing

Calculating the exit conversion in practicalreactors

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Practical reactor performance deviates from thatof ideal reactors :

Packed bed reactor Channeling

CSTR & Batch Dead Zones, Bypass

PFR deviation from plug flow dispersion

Deviation in residence times of molecules

the longitudinal mixing caused by vortices andturbulence

Failure of impellers /mixing devicesHow to design the Practical reactor ??

What design equation to use ??

Approach: (1) Design ideal reactor

(2) Account/correct for deviations 3


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Deviations

In an ideal CSTR, the reactant concentration is uniform

throughout the vessel, while in a real stirred tank, thereactant concentration is relatively high at the pointwhere the feed enters and low in the stagnant regionsthat develop in corners and behind baffles.

In an ideal plug flow reactor, all reactant and productmolecules at any given axial position move at the same ratein the direction of the bulk fluid flow. However, in a realplug flow reactor, fluid velocity profiles, turbulent mixing,

and molecular diffusion cause molecules to move withchanging speeds and in different directions.

The deviations from ideal reactor conditions pose several

problems in the design and analysis of reactors.

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Possible Deviations from ideality:

Short Circuiting or By-Pass Reactant flows into the tank through theinlet and then directly goes out through the outlet without reacting if theinlet and outlet are close by or if there exists an easy route between thetwo.

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1. Dead Zone 2. Short Circuiting

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Three concepts are generally used to describe the

deviations from ideality:

the distribution of residence times (RTD)

the quality of mixing

the model used to describe the system

These concepts are regarded as characteristics ofMixing.

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Analysis of non-ideal reactors is carried out in

three levels:

First Level:

Model the reactors as ideal and account or

correct for the deviationsSecond Level:

Use of macro-mixing information (RTD)

Third Level: Use of micro-mixing information models for

fluid flow behavior10


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RTD Function:

Use of (RTD) in the analysis of non-ideal reactor

performance Mac Mullin & Weber 1935

Dankwerts (1950) organizational structure

Levenspiel & Bischoff, Himmelblau & Bischoff,

Wen & Fan, Shinner

In any reactor there is a distribution of

residence times

RTD effects the performance of the reactor

RTD is a characteristic of the mixing11


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Measurement of RTD

RTD is measured experimentally by injecting an inert

matrerial called tracer at t=0 and measuring itsconcentration at the exit as a function of time.

Injection & Detection points should be very close to

the reactor 12


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ASSUMPTIONS

1. Constant flowrate u(l/min) and fluid density (g/l).

2. Only one flowing phase.

3. Closed system input and output by bulk flow only (i.e.,no diffusion across the system boundaries).

4. Flat velocity profiles at the inlet and outlet.5. Linearity with respect to the tracer analysis, that is,

the magnitude of the response at the outlet isdirectly proportional to the amount of tracer

injected.

6. The tracer is completely conserved within the systemand is identical to the process fluid in its flow andmixing behavior.

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Desirable characteristics of the tracer:

non reactive species

easily detectable

should have physical properties similar to thatof the reacting mixture

completely soluble in the mixture

should not adsorb on the walls

Its molecular diffusivity should be low andshould be conserved

colored and radio active materials are the

most widely used tracers 14


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Types of tracer inputs:

Pulse input

Step input

Ramp input

Sinusoidal inputPulse & Step inputs are most common

Ramp input

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The fraction of material that has spent an amount of

time between t and t+t in the reactor:

dN = C(t) v dt

0

0 )( dttvCN

For pulse input

0

)(N

NttE

0

)(

)()(

dttC

tCtE

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C curve

1)(0

dttE18


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1

0

1

)( ttimeresidenceahavingFractiondttE

t

1

1

)( ttimeresidenceahavingFractiondttE

t

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The age of an element is defined as the time elapsed

since it entered the system.

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12100

.......(22

)(

nn CCCCCh

dttC

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Disadvantages of pulse input

injection must be done in a very short time

when the c-curve has a long tail, the analysis

can give rise to inaccuracies

amount of tracer used should be known

however, require very small amount of tracer

compared to step input

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Step input of tracer

In step input the conc. of tracer is kept at this

level till the outlet conc. equals the inlet conc.

t

out dttECC0

0 )(

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stepC

tC

dt

d

tE

0

)(

)(

For step input:

Disadvantages of Step input:

difficult to maintain a constant tracer conc.

RTD fn requires differentiation can lead

to errors large amount of tracer is required

need not know the amount of tracer used

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Characteristics of the RTD:

E(t) is called the exit age distribution function

or RTD function

describes the amount of time molecules have

spent in the reactor

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Cumulative age distribution function F(t):

t

dttEtF0

)()(

t

dttEtF )()(1

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Relationship between the E and F curves

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C l ti di t ib ti f ti F(t)


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Cumulative age distribution function F(t):

Washout function W(t) = 1 - F(t): 28


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E and F Curves with bypassing

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E and F Curves with Channeling

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0

32/3

3 )()(1 dttEttS m

What is the significance of these moments ??

Moments of RTD:

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If the distribution curve is only known at a number ofdiscrete time values, ti, then the mean residence time isgiven by:

This is what you use in the laboratory

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Variance:

represents the square of the distribution

spread and has the units of (time)2

the greater the value of this moment, thegreater the spread of the RTD

useful for matching experimental curves toone family of theoretical curves

Skewness:

the magnitude of this moment measures theextent that the distribution is skewed inone direction or other in reference to the

mean 37


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Space time vs. Mean residence time:

0

)( dtttEtm0v

V

The Space time and Mean residence time would be

equal if the following two conditions are satisfied:

No density change

No backmixing

In practical reactors the above two may not be valid

and hence there will be a difference between them.

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Normalized RTD function E():

)()( tEE /t

0

)(1 dttE

0

)(1 dE

What is the significance of E() ??

How does E() vs. looks like for two ideal CSTRsof different sizes ??

How does E(t) vs. t looks like for two ideal

CSTRs of different sizes ??

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Using the normalized RTD function, it is possible to

compare the flow performance inside differentreactors.

If E() is used, all perfectly mixed CSTRs havenumerically the same RTD.

If E(t) is used, its numerical values can change fordifferent CSTRs based on their sizes.

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RTD i id l t


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RTD in ideal reactors:

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RTD f id l PFR


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RTD for ideal PFR:

)()( ttE

00)( twhent0)( twhent

1)( dtt

)()()( gdtttg

0

)()( dtttdtttEtm

0

222 0)()()()( dttttdttEtt mm 42

RTD f id l CSTR:


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RTD for ideal CSTR:

0

/ /)( dttedtttEt tm

0

/2

22 )()()(

dtet

dttEtt t

m

Material balance on tracer st to pulse input:

in out = accumulation0 vC = VdC/dt C(t) = C0 e

-t/

/

0

/

0

/

0

0

)(

)()(

t

t

t

e

dteC

eC

dttC

tCtE

eE )(

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RTD for PFR-CSTR series:


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RTD for PFR CSTR series

For a pulse tracer input into CSTR the output

would be : C(t) = C0e-t/s

Then the outlet would be delayed by a time p at the

outlet of the PFR. RTD for the system would be:

pttE 0)(

p

s

t

te

tEsP

/)(

)(

1/s

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If th l f t i i t d d i t th PFR


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If the pulse of tracer is introduced into the PFR,

then the same pulse will appear at the entrance of

the CSTR p seconds later. So the RTD for PFR-CSTR

also would be similar to CSTR-PFR.

Though RTD is same for both, performance is

different

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Remarks:


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Remarks:

RTD is unique for a particular reactor

The reactor system need not be unique for a

given RTD

RTD alone may not be sufficient to analyze the

performance of non-ideal reactors

Along with RTD, a model for the flow behaviour

is required

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Reactor modeling with RTD:


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Reactor modeling with RTD:

I. Zero parameter models:

(a)Segregation model

(b)Maximum mixedness model

II. One parameter models:(a)Tanks-in-series model

(b)Dispersion model

III. Two parameter models:

Micro-mixingmodels

Macro-mixingmodels

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S ti d l (D k ts & Z i t i 1958)


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Segregation model (Dankwerts & Zwietering, 1958)

Characteristics: Flow is visualized in the form of globules

Each globule consists of molecules belonging

to the same residence time

Different globules have different Res. Times

No interaction/mixing between differentglobules

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M i f l b l di b t t d t dt i th t


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Mean conversion of globules spending between t and t+dt in the reactor =

(Conversion achieved after spending a time t in the reactor) X

(Fraction of globules that spend between t and t+dt in the reactor)

dttEtxxd )()(_

0

_

)()( dttEtxx

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Mean conversion in a PFR using Segregation model:


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Mean conversion in a PFR using Segregation model:

Example: A R, I order, Constant density

OrderIforetx kt1)(

00

_

)(1)()1( dttEedttEex ktkt

kktedttex

1)(1

0

_

Mean conversion predicted by Segregation model

matches with ideal PFR

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Mean conversion in a CSTR using Segregation model:


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Mean conversion in a CSTR using Segregation model:


0

/

0

_

/)(1 dteedttEex tktkt

k

kx

1

_

Mean conversion predicted by Segregation model

matches with ideal CSTR

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Mean conversion in a practical reactor using


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Mean conversion in a practical reactor using

Segregation model:


00

_

)(1)()( dttEedttEtxx kt

conduct tracer experiment on the practical reactor

measure C(t) and evaluate E(t)

plot and evaluate mean conversion

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Tanks in series (TIS) Model:


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Tanks in series (TIS) Model:

Material balance on the I reactor for tracer:

V1dC

1/dt = -v C

1 C

1= C

0exp(-t/

1)

Material balance on the II reactor for tracer:

V2 dC2/dt = v C1 v C2 dC2/dt + C2/2 = C0exp(-t/2)258

2/0 ttC

C ittC

CSi il l /2

0


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2/

2

02

teC it

i

eCSimilarly

/

2

03

2

it

i

et

dttC

tCtE

/

3

2

0

3

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2)(

)()(

For n equal sized CSTRs:

it

n

n

en

t

tE

/1

)1()(

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Total = ni = t/ n = t/i


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Total ni = t/ n t/i

nn

t

n

i

n

i e

n

nne

n

tnTEE i

)1(

)(

)1(

)()(1

/1

As the number becomes large,the behavior of the systemapproaches that of PFR

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We can calculate the dimensionless variance 2


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0

2

2

22 )()1(

dE

We can calculate the dimensionless variance

000

2 )()(2)( dEdEdE

1)1(12)1(

)(

0

1

0

12

den

n

den

nn nnn

nn

nnn

nn

n

n

nn

n 11)1(

11

)1(

)1( 22

The number of tanks n = 1/2 = 2/2

If the reaction is I order:n

ik

x)1(

11

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Backmixing or dispersion, is used to represent the combined action of allphenomena namely molecular diffusion turbulent mixing and non


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Ideal Plug flow

phenomena, namely molecular diffusion, turbulent mixing, and non-uniform velocities, which give rise to a distribution of residence times inthe reactor.

If the reactor is an ideal plug flow, the tracer pulse traverses throughthe reactor without distortion and emerges to give the characteristicideal plug flow response. If diffusion occurs, the tracer spreads awayfrom the center of the original pulse in both the upstream anddownstream directions.

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Closed vessel Dispersion Model:


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Closed vessel D spers on Model

Da = Damkohler number = k C0n-1

)1(22

22

2

rPe

rrm

ePePet

2/22/2

2/

)1()1(

41

qPeqPe

Pe

eqeq

qex

PeDq a /41

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The x-axis, labeled macromixing measures the breadth of the residence

time distribution. It is zero for piston flow, fairly broad for the exponential

distribution of a stirred tank, and broader yet for situations involving

bypassing or stagnancy.

The y-axis is micromixing, which varies from none to complete. Micromixing

effects are unimportant for piston flow and have maximum importance forstirred tank reactors.

Well-designed reactors will usually fall in the normal region bounded by the

three apexes, which correspond to piston flow, a perfectly mixed CSTR, and

a completely segregated CSTR.69

Without even measuring the RTD, limits on the performance of most real

t b d t i d b l l ti th f t th th


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reactors can be determined by calculating the performance at the three

apexes of the normal region.

The calculations require knowledge only of the rate constants and the

mean residence time.

When the residence time distribution is known, the uncertainty about

reactor performance is greatly reduced.

A real system must lie somewhere along a vertical line in Normal Region.

The upper point on this line corresponds to maximum mixedness and

usually provides one bound limit on reactor performance.

Whether it is an upper or lower bound depends on the reactionmechanism.

The lower point on the line corresponds to complete segregation and

provides the opposite bound on reactor performance.70


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ANY CLARIFICATIONS ?

Kuhn, Thomas

. . . no theory ever solves all the puzzles with which it is confronted at a

given time; nor are the solutions already achieved often perfect.

9.nin ideal flow

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