DISTRIBUTIONS OF RESIDENCE TIMES FOR CHEMICAL …marcellacroix.espaceweb.usherbrooke.ca/CRE/lecturesM/lecture9M.… · real pfr and real cstr the principal peak accurs at a time smaller

DISTRIBUTIONS OF RESIDENCE DISTRIBUTIONS OF RESIDENCE TIMES FOR CHEMICAL REACTORSTIMES FOR CHEMICAL REACTORS

(9)(9)

Marcel LacroixMarcel LacroixUniversitUniversitéé de Sherbrookede Sherbrooke

DISTRIBUTIONS OF RESIDENCE TIMES FOR DISTRIBUTIONS OF RESIDENCE TIMES FOR CHEMICAL REACTORS:CHEMICAL REACTORS:

INTRODUCTIONINTRODUCTION

• THE REACTORS TREATED THUS FAR- THE PERFECTLY MIXED BATCH, THE PLUG-FLOW TUBULAR, AND THE PERFECTLY MIXED CONTINUOUS TANK REACTORS- HAVE BEEN MODELED AS IDEAL REACTORS.

• WE LIKE THESE FLOW PATTERNS AND IN MOST CASES WE TRY TO DESIGN EQUIPMENT TO APPROACH ONE OF THESE IDEAL REACTORS BECAUSE (1) THEY ARE OFTEN OPTIMUM NO MATTER WHAT WE ARE DESIGNING FOR AND (2) THESE FLOW PATTERNS ARE SIMPLE TO TREAT.

• UNFORTUNALELY, REAL EQUIPMENT DEVIATES FROM THE IDEAL.

M. Lacroix RTD for Chemical Reactors 2

INTRODUCTIONINTRODUCTION

DEVIATION FROM IDEAL FLOW PATTERNS CAN BE CAUSED BY CHANNELING OF FLUID, BY RECYCLING OF FLUID, OR BY CREATION OF STAGNANT REGIONS IN THE VESSEL.


OBJECTIVEOBJECTIVE

• ONE WAY TO ORDER OUR THINKING ON NONIDEAL REACTORS IS TO CONSIDER MODELING THE FLOW PATTERNS IN OUR REACTORS AS EITHER CSTSs OR PFRs AS A FIRSTAPPROXIMATION.

• IN REAL REACTORS, HOWEVER, NONIDEAL FLOW PATTERNS EXIST, RESULTING IN INEFFECTIVE CONTACTING AND LOWER CONVERSIONS THAN IN THE CASE OF IDEAL REACTORS.

• TO ACCOUNT FOR THIS NONIDEALITY, WE USE THE NEXT-HIGHER LEVEL OF APPROXIMATION WHICH INVOLVES THE USE OF MACROMIXING INFORMATION, i.e., THE RESIDENCE-TIME DISTRIBUTION FUNCTION (RTD).

• THE NEXT LEVEL USES INFORMATION AS A MICROSCALE (MICROMIXING) TO MAKE PREDICTIONS ABOUT CONVERSION OF NONIDEAL REACTORS.


RESIDENCERESIDENCE--TIMETIME

• IN AN IDEAL PLUG-FLOW REACTOR, ALL THE ATOMS OF MATERIAL LEAVING THE REACTOR HAVE BEEN INSIDE IT FOR EXACTLY THE SAME AMOUNT OF TIME.

• IN AN IDEAL BATCH REACTOR, ALL THE ATOMS OF MATERIAL WITHIN THE REACTOR HAVE BEEN INSIDE IT FOR AN IDENTICAL LENGTH OF TIME.

• THE TIME THE ATOMS HAVE SPENT IN THE REACTOR IS CALLED THE RESIDENCE TIME OF THE ATOMS IN THE REACTOR.

• THE IDEALIZED PLUG-FLOW AND BATCH REACTORS ARE THE ONLY TWO CLASSES OF REACTORS IN WHICH ALL THE ATOMS IN THE REACTORS HAVE THE SAME RESIDENCE TIME.


RESIDENCERESIDENCE--TIME DISTRIBUTION (RTD)TIME DISTRIBUTION (RTD)• IN A CSTR HOWEVER, SOME OF THE ATOMS ENTERING IT LEAVE

ALMOST IMMEDIATELY (BECAUSE MATERIAL IS BEEING CONTINUOUSLY WITHDRAWN FROM IT) WHILE OTHER ATOMS REMAIN IN THE REACTOR ALMOST FOREVER (BECAUSE ALL THE MATERIAL IS NEVER REMOVED FROM THE REACTOR AT ONE TIME). MANY OF THE ATOMS , OF COURSE, LEAVE THE REACTOR AFTER SPENDING A PERIOD OF TIME SOMEWHERE IN THE VICINITY OF THE MEAN RESIDENCE TIME.

• THUS IN ANY REACTOR THE DISTRIBUTION OF RESIDENCE TIMES CAN SIGNIFICANTLY AFFECT ITS PERFORMANCE.

• THE RESIDENCE-TIME DISTRIBUTION (RTD) OF A REACTOR IS A CHARACTERISTIC OF THE MIXING THAT OCCURS IN THE CHEMICAL REACTOR. IT CHARACTERIZES THE LENGTHS OF TIME VARIOUS ATOMS SPEND AT REACTION CONDITIONS.


MEASUREMENT OF THE RTDMEASUREMENT OF THE RTD

• THE RTD IS DETERMINED EXPERIMENTALLY BY INJECTING AN INERT CHEMICAL, MOLECULE OR ATOM, CALLED A TRACER, INTO THE REACTOR AT SOME TIME t=0 AND THEN MEASURING THE TRACER CONCENTRATION, C, IN THE EFFLUENT STREAM AS A FUNCTION OF TIME.

• COLORED AND RADIOACTIVE MATERIALS ARE THE TWO MOST COMMON TYPES OF TRACERS.

• THE TWO MOST USED METHODS OF INJECTION ARE PULSE INPUT AND STEP INPUT.


MEASUREMENT OF THE RTD: PULSE INPUTMEASUREMENT OF THE RTD: PULSE INPUT


• AN AMOUNT OF TRACER IS SUDDENLY INJECTED IN ONE SHOT INTO THE FEEDSTREAM ENTERING THE REACTOR IN AS SHORT A TIME AS POSSIBLE. THE OUTLET CONCENTRATION IS THEN MEASURED AS A FUNCTION OF TIME.

0N

)(tC


• THE AMOUNT OF TRACER MATERIAL, LEAVING THE REACTOR BETWEEN TIME AND IS THEN

WHERE IS THE EFFLUENT VOLUMETRIC FLOW RATE (m3/s).

• THE RESIDENCE-TIME DISTRIBUTION FUNCTION IS DEFINED AS

• IS A FUNCTION THAT DESCRIBES IN A QUANTITATIVE MANNER HOW MUCH TIME DIFFERENT FLUID ELEMENTS HAVE SPENT IN THE REACTOR.

N∆t tt ∆+

tvtCN ∆=∆ )(

v)(tE

ttEN

tvtCNN

∆=∆

=∆ )()(

00

)(tE



• IF IS NOT KNOWN DIRECTLY, IT CAN BE OBTAINED FROM THE OUTLET CONCENTRATION MEASUREMENTS:

• THE VOLUMETRIC FLOW RATE IS USUALLY CONSTANT; THUS,

• THE FRACTION OF MATERIAL LEAVING THE REACTOR THAT HAS RESIDED IN THE REACTOR FOR TIMES BETWEEN AND IS:

0N

∫=∞

00 )( dttvCN

∫= ∞

0)(

)()(dttC

tCtE

1t

∫2

1

)(t

tdttE

2t


PULSE INPUT: GENERAL COMMENTSPULSE INPUT: GENERAL COMMENTS

• THE PRINCIPAL DIFFICULTIES WITH THE PULSE TECHNIQUE LIE IN THE PROBLEMS CONNECTED WITH OBTAINING A REASONABLE PULSE AT THE REACTOR’S ENTRANCE.

• THE INJECTION MUST TAKE PLACE OVER A PERIOD WHICH IS VERY SHORT COMPARED WITH RESIDENCE TIMES IN VARIOUS SEGMENTS OF THE REACTOR SYSTEM.

• ALSO, THERE MUST BE A NEGLIGIBLE AMOUNT OF DISPERSION BETWEEN THE POINT OF INJECTION AND THE ENTRANCE TO THE REACTOR SYSTEM.

• IF THESE CONDITIONS CAN BE FULFILLED, THIS TECHNIQUE REPRESENTS A SIMPLE AND DIRECT WAY OF OBTAINING THE RESIDENCE-TIME DISTRIBUTION.


EXAMPLE No. 1: PULSE INPUTEXAMPLE No. 1: PULSE INPUT

• A SAMPLE OF THE TRACER HYTANE AT 320 K WAS INJECTED AS A PULSE TO A REACTOR AND THE EFFLUENT CONCENTRATION, MEASURED AS A FUNCTION OF TIME, RESULTED IN THE FOLLOWING DATA:

THE MEASUREMENTS REPRESENT THE EXACT CONCENTRATIONS AT THE TIMESLISTED AND NOT AVERAGE VALUES BETWEEN THE VARIOUS SAMPLING TESTS. (1) CONSTRUCT FIGURES SHOWING C(t) AND E(t) AS FUNCTION OF TIME (t); (2) DETERMINE BOTH THE FRACTION OF MATERIAL LEAVING THE REACTOR THATHAS SPENT BETWEEN 3 AND 6 MINUTES IN THE REACTOR AND THE FRACTION OF MATERIAL LEAVING THAT HAS SPENT BETWEEN 7.75 AND 8.25 MINUTES IN THE REACTOR; (3) 3 MINUTES OR LESS.

t (min) 0 1 2 3 4 5 6 7 8 9 10 12 14

C (g/m3) 0 1 5 8 10 8 6 4 3 2.2 1.5 0.6 0


PULSE INPUT: SOLUTION WITH TRAPEZOIDAL RULEPULSE INPUT: SOLUTION WITH TRAPEZOIDAL RULE

t (min) C(t) (g/m**3) delta t*C(t) E(t) (1/min) delta t*E(t) delta t*E(t)0 0 01 1 0,5 0,02 0,012 5 3 0,1 0,063 8 6,5 0,16 0,134 10 9 0,2 0,18 0,25 8 9 0,16 0,186 6 7 0,12 0,147 4 5 0,08 0,58 3 3,5 0,069 2,2 2,6 0,04410 1,5 1,85 0,0312 0,6 2,1 0,01214 0 0,6 0

50,65

3

0min/50)( mgdttC ⋅≈∫

∞∫6

3)( dttE ∫

3

0)( dttE


MEASUREMENT OF THE RTD: STEP TRACERMEASUREMENT OF THE RTD: STEP TRACER

• THE OUTPUT CONCENTRATION FROM A VESSEL IS RELATED TO THE INPUT CONCENTRATION BY THE CONVOLUTION INTEGRAL:

∫ −=t

inout dttEttCtC0

'' )()()(


MEASUREMENT OF THE RTD: STEP TRACERMEASUREMENT OF THE RTD: STEP TRACER• IN A STEP INPUT, A CONSTANT RATE OF TRACER IS ADDED TO A

FEED AT TIME t=0:

0;)(0;0)(

00

0

>=<=tCtC

ttC


MEASUREMENT OF THE RTD: STEP TRACERMEASUREMENT OF THE RTD: STEP TRACER

• BECAUSE THE INLET CONCENTRATION IS A CONSTANT WITH TIME, , WE CAN TAKE IT OUTSIDE THE INTEGRAL SIGN, i.e.,

• DIVIDING BY YIELDS

• WE DIFFERENTIATE THIS EXPRESSION TO OBTAIN THE RTD FUNCTION:

0C

∫=∫ −=tt

inout dttECdttEttCC0

00

')'(')'()'(

0C

∫ ==⎥⎦

⎤⎢⎣

⎡ '

00

)(')'(t

step

out tFdttECC

stepCtC

dtdtE ⎥

⎦

⎤⎢⎣

⎡=

0

)()(


STEP TRACER: GENERAL COMMENTSSTEP TRACER: GENERAL COMMENTS

• THE STEP METHOD IS EASIER TO CARRY OUT EXPERIMENTALLY THAN THE PULSE TEST.

• THE TOTAL AMOUNT OF TRACER IN THE FEED OVER THE PERIOD OF THE TEST DOES NOT HAVE TO BE KNOWN AS IT DOES FOR THE PULSE TEST.

• IT IS HOWEVER SOMETIMES DIFFUCULT TO MAINTAIN A CONSTANT TRACER CONCENTRATION IN THE FEED.

• OBTAINING THE RTD FROM THIS TEST ALSO INVOLVES DIFFERENTIATION OF THE DATA AND ON OCCASION DIFFERENTIATION OF DATA LEAD TO LARGE ERRORS.

• A LARGE AMOUNT OF TRACER IS REQUIRED FOR THIS TEST. M. Lacroix RTD for Chemical Reactors 17

CHARACTERISTICS OF THE RTD:CHARACTERISTICS OF THE RTD:IDEAL PFR AND IDEAL CSTRIDEAL PFR AND IDEAL CSTR

• SOMETIMES E(t) IS CALLED THE EXIT-AGE DISTRIBUTION. E(t) IS THE AGE DISTRIBUTION OF THE EFFLUENT STREAM. IT CHARACTERIZES THE LENGTHS OF TIME VARIOUS ATOMS SPEND AT REACTION CONDITIONS.

RTD OF A NEARLY IDEAL PFR RTD OF A NEARLY IDEAL CSTR


CHARACTERISTICS OF THE RTD:CHARACTERISTICS OF THE RTD:REAL PFR AND REAL CSTRREAL PFR AND REAL CSTR

THE PRINCIPAL PEAK ACCURS AT A TIME SMALLER THAN THE SPACE TIME (V/v) AND ALSO THAT FLUID EXITS AT A TIME GREATER THAN SPACE-TIME;

PFR

CSTR

THE DEAD ZONE SERVES TO REDUCE THE EFFECTIVE REACTOR VOLUME, INDICATING THAT THE ACTIVE REACTOR VOLUME IS SMALLER THAN EXPECTED.


CHARACTERISTICS OF THE RTD:CHARACTERISTICS OF THE RTD:INTEGRAL RELATIONSHIPSINTEGRAL RELATIONSHIPS

• THE FRACTION OF THE EXIT STREAM THAT HAS RESIDED IN THE REACTOR FOR A PERIOD OF TIME SHORTER THAN A GIVEN VALUE IS

• THE FRACTION OF THE EXIT STREAM THAT HAS RESIDED IN THE REACTOR FOR A PERIOD OF TIME LONGER THAN TIME IS

• F(t) IS THE CUMULATIVE DISTRIBUTION FUNCTION. THE F CURVE IS SOMETIMES USED IN THE SAME MANNER AS THE RTD IN THE MODELING OF CHEMICAL REACTORS.

t

∫=t

dttEtF0

)()(

t

)(1)( tFdttEt

−=∫∞


CHARACTERISTICS OF THE RTD:CHARACTERISTICS OF THE RTD:MEAN RESIDENCE TIMEMEAN RESIDENCE TIME

• IN PREVIOUS LECTURES, WE USED THE SPACE-TIME OR AVERAGE RESIDENCE TIME WHICH IS EQUAL TO

• NO MATTER WHAT RTD EXISTS FOR A PARTICULAR REACTOR, IDEAL OR NONIDEAL, THIS NOMINAL HOLDING TIME IS EQUAL TO THE MEAN RESIDENCE TIME :

τ vV /

τmt

∫=∫

∫=

∞

∞

∞

0

0

0 )()(

)(dtttE

dttE

dtttEtm


CHARACTERISTICS OF THE RTD:CHARACTERISTICS OF THE RTD:VARIANCEVARIANCE

• ANOTHER IMPORTANT PARAMETER IN COMPARING RTDs IS THE VARIANCE. IT IS DEFINED AS

• THE MAGNITUDE OF THIS PARAMETER IS AN INDICATION OF THE SPREAD OF THE DISTRIBUTION.

• THE VARIANCE WILL BE USED LATER IN ONE-PARAMETER MODELS.

∫ −=∞

0

22 )()( dttEtt mσ


EXAMPLE No. 2:EXAMPLE No. 2:MEAN RESIDENCE TIME AND VARIANCEMEAN RESIDENCE TIME AND VARIANCE

t (min) C(t) (g/m**3)C (min*g/m**3) E(t) (1/min) t*E(t) dt*t*E(t) t-tm (t-tm)**2*E(t)dt*(t-tm)**2*E(t)0 0 0 0 -5,19 01 1 0,5 0,02 0,02 0,01 -4,19 0,351122 0,1755612 5 3 0,1 0,2 0,11 -3,19 1,01761 0,6843663 8 6,5 0,16 0,48 0,34 -2,19 0,767376 0,8924934 10 9 0,2 0,8 0,64 -1,19 0,28322 0,5252985 8 9 0,16 0,8 0,8 -0,19 0,005776 0,1444986 6 7 0,12 0,72 0,76 0,81 0,078732 0,0422547 4 5 0,08 0,56 0,64 1,81 0,262088 0,170418 3 3,5 0,06 0,48 0,52 2,81 0,473766 0,3679279 2,2 2,6 0,044 0,396 0,438 3,81 0,6387084 0,556237210 1,5 1,85 0,03 0,3 0,348 4,81 0,694083 0,666395712 0,6 2,1 0,012 0,144 0,444 6,81 0,5565132 1,250596214 0 0,6 0 0 0,144 8,81 0 0,5565132

50,65 5,194 6,0325493

3

0min/50)( mgdttC ⋅≈∫

∞

∫=∞

0)( dtttEtm ∫ −=

∞

0

22 )()( dttEtt mσ

INTEGRATION IS CARRIED OUT WITH THE TRAPEZOIDAL RULE


RTDs IN BATCH AND PLUGRTDs IN BATCH AND PLUG--FLOW REACTORSFLOW REACTORS

• THE RTDs IN THESE IDEAL REACTORS ARE THE SIMPLEST TO CONSIDER. ALL THE ATOMS LEAVING SUCH REACTORS HAVE SPENT PRECISELY THE SAME AMOUNT OF TIME WITHIN THE REACTORS. THE DISTRIBUTION FUNCTION IN SUCH A CASE IS A SPIKE OF INFINITE HEIGHT AND ZERO WIDTH, WHOSE AREA IS EQUAL TO 1. THE SPIKE OCCURS AT τ== vVt /

)()( τδ −= ttE

∫ =−∫ =

=∞=≠=∞

∞−

∞

∞−);()()(;1)(

;0;)(;0;0)(

ττδδ

δδ

gdxxxgdxx

xxxx

DIRAC DELTA FUNCTION:

PROPERTIES:


RTDs IN SINGLERTDs IN SINGLE--CSTRsCSTRs

• IN AN IDEAL CSTR THE CONCENTRATION OF ANY SUBSTANCE IN THE EFFLUENT STREAM IS IDENTICAL TO THE CONCENTRATION THROUGHOUT THE REACTOR.

• CONSIDER AN INERT TRACER THAT HAS BEEN INJECTED AS A PULSE AT TIME t = 0 INTO A CSTR. THE MATERIAL BALANCE FOR t > 0 YIELDS

• THE SOLUTION TO THIS EQUATION ISWHERE . THEN, THE RTD FUNCTION BECOMES

dtdCVvC =−0

)/exp()( 0 τtCtC −=

ττ

τ

τ )/exp(

)/exp(

)/exp(

)(

)()(

00

0

0

t

dttC

tC

dttC

tCtE −=

∫ −

−=

∫= ∞∞

vV /=τ

IN – OUT = ACCUMULATION


REACTOR MODELING WITH THE RTD:REACTOR MODELING WITH THE RTD:CLASSIFICATION OF MODELSCLASSIFICATION OF MODELS

• WE NOW USE THE RTD TO PREDICT CONVERSION IN A REAL REACTOR.

• THE REACTOR MODELS ARE CLASSIFIED ACCORDING TO THE NUMBER OF ADJUSTABLE PARAMETERS:

1. ZERO ADJUSTABLE PARAMETERS*SEGREGATION MODEL*MAXIMUM MIXEDNESS MODEL

2. ONE ADJUSTABLE PARAMETER*TANKS-IN-SERIES MODEL*DISPERSION MODEL


REACTOR MODELING WITH THE RTD:REACTOR MODELING WITH THE RTD:FIRSTFIRST--ORDER REACTIONSORDER REACTIONS

• THE RTD TELLS US HOW LONG THE VARIOUS FLUID ELEMENTS HAVE BEEN IN THE REACTOR. IT DOES NOT TELL US ANYTHING ABOUT THE EXCHANGE OF MATTER BETWEEN THE FLUID ELEMENTS (THE MIXING).

• FOR FIRST-ORDER REACTIONS THE CONVERSION IS INDEPENDENT OF CONCENTRATION:

KNOWLEDGE OF THE LENGTH OF TIME EACH MOLECULE SPENDS IN THE REACTOR IS ALL THAT IS NEEDED TO PREDICT CONVERSION.

• THEREFORE, ONCE THE RTD IS DETERMINED WE CAN PREDICT THE CONVERSION THAT WILL BE ACHIEVED IN THE REAL REACTOR PROVIDED THAT THE SPECIFIC REACTION RATE FOR THE FIRST-ORDER REACTION IS KNOWN.

)1( XkdtdX

−=


REACTOR MODELING WITH THE RTD:REACTOR MODELING WITH THE RTD:REACTIONS OTHER THAN FIRSTREACTIONS OTHER THAN FIRST--ORDERORDER

• FOR REACTIONS OTHER THAN FIRST-ORDER, KNOWLEDGE OF THE RTD IS NOT SUFFICIENT TO PREDICT CONVERSION. IN THESE CASES, THE DEGREE OF MIXING OF MOLECULES MUST BE KNOWN IN ADDITION TO HOW LONG EACH MOLECULE SPENDS IN THE REACTOR. CONSEQUENTLY, WE MUST DEVELOP MODELS THAT ACCOUNT FOR THE MIXING OF MOLECULES INSIDE THE REACTOR.

• THE MORE COMPLEX MODELS OF NONIDEAL REACTORS NECESSARY TO DESCRIBE REACTIONS OTHER THAN FIRST-ORDER MUST CONTAIN INFORMATION ABOUT MICROMIXING IN ADDITION TO THAT OF MACROMIXING.


REACTOR MODELING WITH THE RTD:REACTOR MODELING WITH THE RTD:MACROMIXING AND MICROMIXINGMACROMIXING AND MICROMIXING

• MACROMIXING PRODUCES A DISTRIBUTION OF RESIDENCE TIMES WITHOUT, HOWEVER, SPECIFYING HOW MOLECULES OF DIFFERENT AGES ENCOUNTER ONE ANOTHER IN THE REACTOR.

• MICROMIXING DESCRIBES HOW MOLECULES OF DIFFERENT AGES ENCOUNTER ONE ANOTHER IN THE REACTOR. THERE ARE TWO EXTREMES OF MICROMIXING:

1. COMPLETE SEGREGATION: ALL MOLECULES OF THE SAME AGE GROUP REMAIN TOGETHER AS THEY TRAVEL THROUGH THE REACTOR AND ARE NOT MIXED UNTIL THEY EXIT IT.

2. COMPLETE MIXING: MOLECULES OF DIFFERENT AGE GROUPS ARE COMPLETELY MIXED AT THE MOLECULAR LEVEL AS SOON AS THEY ENTER THE REACTOR.


REACTOR MODELING WITH THE RTD:REACTOR MODELING WITH THE RTD:MACROFLUID AND MICROFLUIDMACROFLUID AND MICROFLUID

MACROFLUID MICROFLUID

• MACROFLUID: A FLUID IN WHICH THE GLOBULES OF A GIVEN AGE DO NOT MIX WITH OTHER GLOBULES (LATE MIXING OR SEGREGATION).

• MICROFLUID: A FLUID IN WHICH MOLECULES ARE FREE TO MOVE EVERYWHERE (EARLY MIXING OR MAXIMUM MIXEDNESS).


ZEROZERO--PARAMETER MODEL:PARAMETER MODEL:SEGREGATION MODEL (CSTR)SEGREGATION MODEL (CSTR)

IN THE SEGREGATED FLOW MODEL, THE FLOW THROUH THE CSTR CONSISTS OF A CONTINUOUS SERIES OF GLOBULES THAT RETAIN THEIR IDENTITY DURING THEIR STAY IN THE REACTOR.

THE SEGRAGATED MODEL LUMPS ALL THE MOLECULES THAT HAVE THE SAME RESIDENCE TIME IN THE REACTOR INTO THE SAME GLOBULES.


ZEROZERO--PARAMETER MODEL:PARAMETER MODEL:SEGREGATION MODEL (PFR)SEGREGATION MODEL (PFR)

• A WAY OF LOOKING AT THE SEGREGATION MODEL FOR A CONTINUOUS-FLOW SYSTEM IS A PFR IN WHICH BATCHES OF MOLECULES ARE REMOVED FROM THE REACTOR AT DIFFERENT LOCATIONS ALONG THE REACTOR IN SUCH A MANNER SO AS TO DUPLICATE THE RTD FUNCTION.


SEGREGATION MODEL:SEGREGATION MODEL:MEAN CONVERSION MEAN CONVERSION

• TO DETERMINE THE MEAN CONVERSION IN THE EFFLUENT STREAM, WE MUST AVERAGE THE CONVERSIONS OF VARIOUS GLOBULES IN THE EXIT STREAM:

• SUMMING OVER ALL GLOBULES, THE MEAN CONVERSION IS:

• CONSEQUENTLY, IF WE HAVE THE BATCH REACTOR EQUATION X(t) AND MEASURE THE RTD EXPERIMENTALLY, WE CAN FIND THE MEAN CONVERSION IN THE EXIT STREAM.

dttEtXXd )()( ⋅=MEAN CONVERSION OF THOSE GLOBULES SPENDING BETWEEN t AND t+delta t IN REACTOR

CONVERSION ACHIEVED AFTER SPENDING A TIME t IN REACTOR

FRACTION OF GLOBULES THAT SPEND BETWEEN t AND t+delta t IN REACTOR

∫=∞

0)()( dttEtXX


SEGREGATION MODEL: MEAN CONVERSION FOR SEGREGATION MODEL: MEAN CONVERSION FOR FIRSTFIRST--ORDER REACTION IN BATCH REACTORORDER REACTION IN BATCH REACTOR

• CONSIDER THE FIRST-ORDER REACTION:

• FOR A BATCH REACTOR,

• FOR CONSTANT VOLUME AND WITH , WE GET:

• THE SOLUTION IS

• THE MEAN CONVERSION IS THEN

productsA k⎯→⎯

VrdtdN

AA −=

−

)1(0 XNN AA −=

)1( XkdtdX

−=

)exp(1)( kttX −−=

∫ ⋅−−∫ =⋅−−∫ =⋅=∞∞∞

000)()exp(1)())exp(1()()( dttEktdttEktdttEtXX


EXAMPLE No. 3:EXAMPLE No. 3:SEGREGATION MODEL FOR MEAN CONVERSION IN AN SEGREGATION MODEL FOR MEAN CONVERSION IN AN

IDEAL PFR IDEAL PFR • OBTAIN THE MEAN CONVERSION OF A FIRST-ORDER

REACTION USING THE SEGREGATION MODEL WHEN THE RTD IS EQUIVALENT TO AN IDEAL PFR. COMPARE THE RESULTS WITH THOSE OBTAINED FROM THE DESIGN EQUATIONS.

• SOLUTION FOR AN IDEAL PFRTHE RTD FUNCTION IS . THUS, THE MEAN CONVERSION IS

USING THE INTEGRAL PROPERTIES OF THE DIRAC DELTA FUNCTION, WE OBTAIN . RECALL THAT FOR A PFR AFTER COMBINING THE MOLE BALANCE, RATE LAW, AND STOICHIOMETRIC RELATIONSHIPS,

WHICH YIELDS

∫ −−−=∫ −−=∫=∞∞∞

000)()exp(1)()exp(1)()( dttktdttEktdttEtXX τδ

)()( τδ −= ttE

)exp(1 τkX −−=

)1( XkddX

−=τ

)exp(1 τkX −−=


EXAMPLE No. 4: SEGREGATION MODEL FOREXAMPLE No. 4: SEGREGATION MODEL FORMEAN CONVERSION IN AN IDEAL CSTR MEAN CONVERSION IN AN IDEAL CSTR

• OBTAIN THE MEAN CONVERSION OF A FIRST-ORDER REACTION USING THE SEGREGATION MODEL WHEN THE RTD IS EQUIVALENT TO AN IDEAL CSTR. COMPARE THE RESULTS WITH THOSE OBTAINED FROM THE DESIGN EQUATIONS.

• SOLUTION FOR AN IDEAL CSTRTHE RTD FUNCTION IS . THUS, THE MEAN CONVERSION IS

PERFORMING THE INTEGRATION, WE GET

COMBINING THE CSTR MOLE BALANCE, THE RATE LAW, AND STOICHIOMETRY, WE HAVE THUS,

∫+−

−=∫ −−=∫=∞∞∞

000

])/1(exp[1)()exp(1)()( dttkdttEktdttEtXXτ

τττ /)/exp()( ttE −=

kkXτ

τ+

=1

VXkCXCvANDVrXF AAAA )1(__ 0000 −=−=

kkXτ

τ+

=1


EXAMPLE No. 5: SEGREGATION MODEL FOREXAMPLE No. 5: SEGREGATION MODEL FORMEAN CONVERSION IN A REAL REACTOR MEAN CONVERSION IN A REAL REACTOR


• CALCULATE THE MEAN CONVERSION IN THE REACTOR THAT WE HAVE CHARACTERIZED BY RTD MEASUREMENTS IN THE PREVIOUS EXAMPLE FOR A FIRST-ORDER LIQUID-PHASE IRREVERSIBLE REACTION IN A COMPLETELY SEGREGATED FLUID:

• THE SPECIFIC REACTION RATE IS k=0.1 min-1 AT 320 K.• BECAUSE EACH GLOBULE ACTS AS A BATCH REACTOR OF

CONSTANT VOLUME, WE USE THE BATCH REACTOR DESIGN EQUATION TO ARRIVE AT THE EQUATION GIVING CONVERSION AS A FUNCTION OF TIME:

• THE MEAN CONVERSION IS OBTAINED FROM THE INTEGRAL

productsA k⎯→⎯

)1.0exp(1)exp(1 tktX −−=−−=

∫=∞

0)()( dttEtXX

EXAMPLE No. 5: SEGREGATION MODEL FOREXAMPLE No. 5: SEGREGATION MODEL FORMEAN CONVERSION IN A REAL REACTOR MEAN CONVERSION IN A REAL REACTOR

• THE MEAN CONVERSION IS THE AREA UNDER THE CURVE X(t)*E(t). THE RESUTL IS

t(min) E(t) min-1 X(t) X(t)*E(t) min-1

0 0.000 0.000 0.0000

1 0.020 0.095 0.0019

2 0.100 0.181 0.0180

3 0.160 0.259 0.0414

4 0.200 0.330 0.0660

5 0.160 0.393 0.0629

6 0.120 0.451 0.0541

7 0.080 0.503 0.0402

8 0.060 0.551 0.0331

9 0.044 0.593 0.0261

10 0.030 0.632 0.01896

12 0.012 0.699 0.0084

14 0.000 0.750 0.0000

385.0=X


EXAMPLE No.5: SEGREGATION MODEL FOR A REAL EXAMPLE No.5: SEGREGATION MODEL FOR A REAL REACTOR: SOLUTION WITH AN ODE SOLVERREACTOR: SOLUTION WITH AN ODE SOLVER

• THE PREVIOUS EXAMPLE CAN BE SOLVED WITH AN ORDINARY DIFFERENTIAL EQUATION SOLVER:

1. E(t) IS FIRST CURVED FITTED WITH A POLYNOMIAL.

2. THE FOLLOWING SET OF COUPLED DIFFERENTIAL EQUATIONS IS NEXT SOLVED:

AND

WITH . THE RATE OF REACTION IS EXPRESSED AS A FUNCTION OF CONVERSION.

)()( tEtXdtXd=

0A

A

Cr

dtdX −

=

0)0()0( == XX Ar−


ZEROZERO--PARAMETER MODEL:PARAMETER MODEL:MAXIMUM MIXEDNESSMAXIMUM MIXEDNESS

• IN A REACTOR WITH A SEGREGATED FLUID, MIXING BETWEEN PARTICLES OF FLUID DOES NOT OCCUR UNTIL THE FLUID LEAVES THE REACTOR (MINIMUM MIXEDNESS).

• WE NOW CONSIDER THE OTHER EXTREME, THAT OF MAXIMUM MIXEDNESS.

AS SOON THE FLUID ENTERS THE REACTOR (THROUGH THE SIDE HOLES) IT IS COMPLETELY MIXED RADIALLY. THE GLOBULES AT THE FAR LEFT CORRESPOND TO THE MOLECULES THAT SPEND A LONG TIME IN THE REACTOR WHILE THOSE AT THE FAR RIGHT CORRESPOND TO THE MOLECULES THAT CHANNEL THROUGH THE REACTOR.


MAXIMUM MIXEDNESS MODELMAXIMUM MIXEDNESS MODEL

• IS THE TIME TAKEN BY THE FLUID TO MOVE FROM A PARICULAR POINT TO THE END OF THE REATOR. IT IS THE LIFE EXPECTANCY OF THE FLUID IN THE REACTOR AT THAT POINT.

λ


MAXIMUM MIXEDNESS MODELMAXIMUM MIXEDNESS MODEL

• THE VOLUMETRIC FLOW RATE OF FLUID FED INTO THE SIDE OF THE REACTOR IN THE INTERVAL CORRESPONDING TO THAT BETWEEN AND IS

• THE VOLUMETRIC FLOW RATE INSIDE THE REACTOR AT IS

• THE VOLUME OF FLUID WITH A LIFE EXPECTANCY BETWEENAND IS

• THE RATE OF GENERATION OF THE SUBSTANCE A IN THIS VOLUME IS

λλ ∆+ λ λλ ∆)(0Ev

λ

∫ −==∞

λλλλλ ))(1()()( 00 FvdEvv

λλ ∆+λλλ ∆−=∆ ))(1(0 FvV

λλ ∆−=∆ ))(1(0 FvrVr AA


MAXIMUM MIXEDNESS MODEL: CONVERSIONMAXIMUM MIXEDNESS MODEL: CONVERSION

• A BALANCE ON SUBSTANCE A BETWEEN ANDTHEN YIELDS

• WE MAY REWRITE THIS EQUATION IN TERMS OF CONVERSION

• THE BOUNDARY CONDITION IS

λ λλ ∆+

)(1)()( 0 λ

λλ F

ECCrd

dCAAA

A

−−+−=

XF

ECr

ddX

A

A ⋅−

+=)(1

)(0 λ

λλ

0__, 0 ==∞→ XorCC AAλ


MAXIMUM MIXEDNESS MODEL: SOLUTIONMAXIMUM MIXEDNESS MODEL: SOLUTION

• BECAUSE ‘ODE’ SOFTWARE PACKAGES DO NOT INTEGRATE BACWARDS, WE NEED FIRST TO CHANGE THE INDEPENDENT VARIABLE IN THE CONVERSION EQUATION.

• WE SET WHERE IS THE LONGEST TIME MEASURED IN THE E(t) CURVE. THUS, THE EQUATION TO INTEGRATE BECOMES

• IT IS INTEGRATED BETWEEN THE LIMIT ANDTO FIND THE EXIT CONVERSION AT (WHICH CORRESPONDS TO ).

λ

λ−=Tz T

XzTF

zTECr

dzdX

A

A ⋅−−

−−−=

)(1)(

0

0=z Tz =Tz =

0=λ


EXAMPLE No. 6:EXAMPLE No. 6:CONVERSION BOUNDS FOR NONIDEAL REACTOR:CONVERSION BOUNDS FOR NONIDEAL REACTOR:

• THE LIQUID-PHASE, SECOND-ORDER DIMERIZATION

FOR WHICH IS CARRIED OUT AT A REACTION TEMPERATURE OF 320K. THE FEED IS PURE A WITH . THE REACTOR IS NONIDEAL AND PERHAPS COULD BE MODELED AS A CSTR. THE REACTOR VOLUME IS 1000 dm3 AND THE FEED RATE IS 25 dm3/min. A TRACER TEST WAS RUN ON THIS REACTOR AND THE RESULTS ARE GIVEN IN COLUMNS 1 AND 2 OF THE FOLLOWING TABLE. WE WISH TO KNOW THE BOUNDS ON THE CONVERSION FOR DIFFERENT POSSIBLE DEGREES OF MICROMIXING FOR THE RTD OF THIS REACTOR. WHAT ARE THESE BOUNDS?

2;2 AA kCrBA −=→min/01.0 3 ⋅= moledmk

30 /8 dmmoleCA =


EXAMPLE No.6: SEGREGATION MODEL EXAMPLE No.6: SEGREGATION MODEL

• THE BATCH REACTOR EQUATION FOR A SECOND-ORDER REACTION IS

• THE CONVERSION FOR A COMPLETELY SEGREGATED FLUID IN REACTOR ISt

ttkC

tkCXA

A

08.0108.0

1 0

0

+=

+=

∫=∞

0)()( dttEtXX NUMERICAL INTEGRATION CARRIED

OUT WITH SIMPLE TRAPEZOID RULE

CONVERSION IF FLUID SEGREGATED: 61%

time (min) C(mg/dm**3)E(t) (min**-1) X(t) X(t)*E(t) X(t)*E(t)*delt0 112 0,028 0 0 05 95,8 0,024 0,286 0,00686 0,0171610 82,2 0,0206 0,444 0,00915 0,0400315 70,6 0,0177 0,545 0,00965 0,0469820 60,9 0,0152 0,615 0,00935 0,0474930 45,6 0,0114 0,706 0,00805 0,0869840 34,5 0,00863 0,762 0,00658 0,0731250 26,3 0,00658 0,8 0,00526 0,0592070 15,7 0,00393 0,848 0,00333 0,08597

100 7,67 0,00192 0,889 0,00171 0,07559150 2,55 0,000638 0,923 0,00059 0,05739200 0,9 0,000225 0,941 0,00021 0,02001

0,60993


MAXIMUM MIXEDNESS MODEL MAXIMUM MIXEDNESS MODEL

• THE EQUATION TO BE SOLVED IS

• PARAMETERS:

• INITIAL CONDITIONS:

• CURVE FITTING WITH POLYMATH/POLYNOMIAL REGRESSION:

AND F IS OBTAINED FROM INTEGRATION OF THAT IS

XzTF

zTECr

dzdX

A

A ⋅−−

−−−=

)(1)(

0

;8);1(;01.0;;200 002 =−==−== AAAAA CXCCkkCrT

;999.0,0),200(0 ==== FXz λ

44

33

2210)( λλλλλ aaaaaE ++++=

)(λλ

EddF

=

5432)(

54

43

32

21

0λλλλλλ aaaaaF ++++=


MAXIMUM MIXEDNESS MODEL:MAXIMUM MIXEDNESS MODEL:CURVE FITTING FOR E(t)CURVE FITTING FOR E(t)

• THE BEST FIT FOR : TWO POLYNOMIALS )(λE

E(t)

0

0,005

0,01

0,015

0,02

0,025

0,03

0 50 100 150 200 250TIME (min)

E(t)

(1/m

in)

0.028 -8.657x10-4 1.353x10-5 -1.180x10-7 4.447x10-10

0.015 -2.407x10-4 1.362x10-6 -2.640x10-9 0.0

70<λ

70>λ

0a 1a 2a 3a 4a


MAXIMUM MIXEDNESS MODEL:MAXIMUM MIXEDNESS MODEL:POLYMATH COMPUTER PROGRAMPOLYMATH COMPUTER PROGRAM


• ODE Report (RKF45)

• Differential equations as entered by the user• [1] d(x)/d(z) = -(ra/cao+E/(1-F)*x)

• Explicit equations as entered by the user• [1] cao = 8• [2] lam = 200-z• [3] E2 = -264e-11*lam^3+13618e-10*lam^2-2407e-07*lam+15e-03• [4] ca = cao*(1-x)• [5] E1 = 444658e-15*lam^4-11802e-11*lam^3+135358e-10*lam^2-8657e-07*lam+28004e-06• [6] E = if (lam<=70) then (E1) else (E2)• [7] F1 = 444658e-15/5*lam^5-11802e-11/4*lam^4+135358e-10/3*lam^3-865652e-09/2*lam^2+28004e-

06*lam• [8] F2 = -(-930769e-11*lam^3+502846e-10*lam^2-941e-05*lam+618231e-06-1)• [9] k = 1E-02• [10] ra = -k*ca^2• [11] F = if (lam<=70) then (F1) else (F2)• [12] EF = E/(1-F)

• Independent variable • variable name : z• initial value : 0• final value : 200

MAXIMUM MIXEDNESS MODEL: RESULTMAXIMUM MIXEDNESS MODEL: RESULT

CONVERSION FOR FLUID WITH MAXIMUM MIXEDNESS: 56%


CONVERSION BOUNDS FOR NONIDEAL REACTOR: CONVERSION BOUNDS FOR NONIDEAL REACTOR: CONCLUSIONCONCLUSION

• THE CONVERSION FOR A CONDITION OF COMPLETE SEGREGATION IS 61%.

• THE CONVERSION FOR A CONDITION OF MAXIMUM MIXEDNESS IS 56%.

• THERE IS LITTLE DIFFERENCE IN THE CONVERSIONS FOR THE TWO CONDITIONS. WITH BOUNDS THIS NARROW, THERE WOULD NOT BE MUCH POINT IN MODELING THE REACTOR TO IMPROVE THE PREDICTABILITY OF CONVERSION.


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