ert 208/4 reaction engineering: distribution of residence times for reactors (part b) by; mrs hafiza...

ERT 208/4 REACTION

ENGINEERING: Distribution of

Residence Times for Reactors

(PART B) By; Mrs Hafiza Binti Shukor

ERT 208/4 REACTION ENGINEERINGSEM 2 (2009/2010)


C

C C

t t

Pulse injection

Step injection

Pulse response

Step response

MEASUREMENT OF MEASUREMENT OF RTD…RTD…2 most used methods of

injection :A) pulse inputB) step input

Step Input / Step Tracer Step Input / Step Tracer Experiment …Experiment …


Formulated more general relationship between a time varying tracer injection and the corresponding conc in the effluent.

We shall state without development that the output conc from a vessel is related to the input conc by the convolution integral;

t

inout dttEttCtC0

)'()'()(



Analyze a step input in the tracer conc for a system with a constant volumetric flowrate.

Consider a constant rate of tracer addition to a feed that is initiated at time t=0. Before this time, no tracer was added to the feed. Symbolically, we have

)(tCo

0 t<0(Co) constant t>0



The conc of tracer in the feed to the reactor is kept at this level until the conc in the effluent is indistinguishable from that in the feed; the test may then be discontinued.

CinCout

t t

Step injection Step response

Because the inlet conc is a constant with time, Co we can take it outside the integral sign, that is

t

oout dttECC0

)'(



Dividing by Co yields,

')'(0t

oout dttECC

)(')'(0

tFdttEC

C t

stepo

out

Differentiate this expression to obtain RTD function of E(t),

stepoC

tC

dt

dtE

)()(



The +ve step is usually easier to carry out experimentally than the pulse test, and it has the additional advantage that the total amount of tracer in the feed over the period of the test does not have to be known as it does in the pulse test.

Disadvantages for step input method;a)Sometimes difficult to maintain a constant tracer conc in the feed.b)Differentiation of the data (lead to large errors)c)Required large amount of tracer (expensive)

Characteristics of the Characteristics of the RTD…RTD…


E(t) sometimes is called as exit-age distribution function. It characterizes the lengths of time various atoms spend at reaction conditions.

RTD for Plug Flow ReactorRTD for Near Perfectly Mixed CSTR

RTD that commonly observed



RTD for Packed-Bed Reactor with Dead Zones & Channeling

Dead zones – serve to reduce the effective reactor volume, indicating that the active reactor volume is smaller than expected.



CSTR with dead zones Tank reactor with short-circuting flow (bypass)



Integral RelationshipThe fraction of the exit stream that has resided in the

reactor for a period of time shorter than a given value t is equal to the sum over all times less than t of E(t)∆t, or expressed continuously,

)(

____

____

__

)(0

tF

ttimethanlessfor

reactorinbeenhaswhich

effluentoffraction

dttEt

)(1

____

____

__

)( tF

ttimethanlongerfor


effluentoffraction

dttEt

Analogously,



Cumulative distribution function and called it F(t).

)(

____

____

__

)(0

tF

ttimethanlessfor


effluentoffraction

dttEt

Can calculate F(t) at various time t from the area under the curve of an E(t)

vs t plot.



The shape of the F(t) curve is shown for a tracer response to a step input in figure below..

F(t)

t

1.0

0.8

40

Cumulative distribution curve

80% [F(t)] of the molecules spend 40 min or less in the reactor and 20% [1-F(t)] of the molecules spend longer than 40min in the reactor

Mean Residence Mean Residence Time…Time…


Ideal Reactor – parameter frequently used was SPACE TIME @ AVERAGE RESIDENCE TIME, .

Ideal @ Non ideal Reactor – this nominal holding time, is equal to mean residence time,

The mean value of variable is equal to the first moment of the RTD function, E(t). Thus, the mean residence time is,

mt

0

0

0 )()(

)(dtttE

dttE

dtttEtm

mt

Variance…Variance…


Variance @ square of the standard deviation.

Is defined by,

Is indication of spread of the distribution (greater value of variance, the greater distribution’s spread)

2

0

22 )()( dttEtt m

2

Skewness…Skewness…


Is defined by,

Measures the extent that a distribution is skewed in one direction @ another in reference to the mean.

3s

0

3

2/3

3 )()(1

dttEtts m

Example: Mean Example: Mean Residence Time & Residence Time &

Variance Variance Calculations Calculations


Calculate the mean residence time and the variance for the reactor characterized in previous example by the RTD obtained from a pulse input at 320K.

Solution;The mean residence time,

The area under the curve of plot of tE(t) as a function of t will yield tm.

0

)( dtttEtm

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


To calculate tm we have to used integration formula in Appendix A.4 (text book)using tE(t) data to get area under the curve of tE(t) VS t

min15.5mt

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

C 0 1 5 8 10 8 6 4 3 2.2 1.5 0.6 0

E(t) 0 0.02

0.1 0.16 0.2 0.16 0.12 0.08 0.06 0.044

0.03 0.012

0

tE(t)

t-tm

(t-tm)2E(t)

10

0

14

100)()()( dtttEdtttEdtttEtm


To calculate variance, we use equation,

Once u finished calculate this data for time 0 to 14min, we have to used integration formula to get variance value.

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

C 0 1 5 8 10 8 6 4 3 2.2 1.5 0.6 0

E(t) 0 0.02

0.1 0.16 0.2 0.16 0.12 0.08 0.06 0.044

0.03 0.012

0

tE(t)

t-tm

(t-tm)2E(t)

0

22 )()( dttEtt m

14

10

210

0

2

0

22 )()()()()()( dttEttdttEttdttEtt mmm

min4.2

min1.6 22

Internal Age Internal Age Distribution, I(t)…Distribution, I(t)…


Is defined by,

Is a fuction such that fraction of material inside the reactor.

It characterizes the time the material has been (and still is) in the reactor at a particular time.

3s

t

dttEtFtI0

)(11

)](1[1

)(

RTD in Batch & Plug-Flow RTD in Batch & Plug-Flow ReactorsReactors


In Batch & PFR, all the atoms leaving such reactors have spent precisely the same amount of time within the reactors.

The distribution function in such case is a spike of infinite height & zero width, whose area is equal to 1.

The spike occurs at or

Mathematically, this spike is represented by the Dirac delta function: )()( ttE

/Vt )()( ttE t



The Dirac delta function has the following properties:

dtttdtttEtm )()(0

)(x0 when x=0

∞ when x=0

Mean residence time is,

Variance is,

0)()(0

22

dttt



dtttFt

)()(0

The cumulative distribution function F(t) is,

E(t)

t

in

PFR response to a pulse tracer input

out

F(t)

1.0

RTD in Single CSTRRTD in Single CSTR


In an ideal CSTR, the conc of any substances in the effluent stream is identical to the conc throughout the reactor.

Use tracer balance to determine RTD for CSTR.

E(t) for CSTR,

eE )(

/

0

/

/

0)(

)()(

t

t

o

t

o e

dteC

eC

dttC

tCtE

t

Where,

RTD for CSTRRTD for CSTR


edEF 1)()(0

The cumulative distribution function is,

CSTR response to a pulse tracer input

1.0

)(F)(E

RTD for CSTRRTD for CSTR


dtet

dtttEt t

m 0

/

0)(

Mean residence time is,

Variance is,

0

222/

0

22 )1(

)( dxexdte

t xt

End For Part B


ert 208/4 reaction engineering: distribution of residence times for reactors (part b) by; mrs hafiza...

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inlet conc