Ch E 441 RTDs

8/3/2019 Ch E 441 RTDs

1/35

Ch E 441 - Chemical Kinetics and Reaction Engineering

Residence Time Distributions

in Chemical Reactors
http://www.bbc.co.uk/northernireland/yourplaceandmine/images/helpabout/keyboard_250x145.jpg

8/3/2019 Ch E 441 RTDs

2/35


The assumption of a perfectly mixed reactoroften falls short of reality.

Residence time distributions are used to model

the imperfect mixing behavior of real reactors.

Cumulative age, F(t)

External age, E(t)

Internal age, I(t)

8/3/2019 Ch E 441 RTDs

3/35


Gas-liquid CSTR (A(g) + B(l)

C(l)) Reaction occurs at gas-liquid interface

Liquid phase is perfectly mixed

Rate is proportional to bubble surface area

Residence time of gas bubble in reactor is

proportional to bubble volume

Larger bubble escape rapidly

Smaller bubbles may remain in reactor until consumed

Understanding of RTDs is necessary for analysis

8/3/2019 Ch E 441 RTDs

4/35


PBR Sections of the catalyst bed may offer less resistance

to flow, resulting in a preferred pathway through the

bed.

Molecules flowing through the channel do not

spend as much time in the PBR as those taking

another path.

Consequently, there is a distribution of residencetime for the PBR.

8/3/2019 Ch E 441 RTDs

5/35


CSTR Short-circuiting may occur (the direct movement of

material from inlet to outlet.

Dead zones may exist (regions with a minimum of

mixing and thus virtually no reaction takes place).

8/3/2019 Ch E 441 RTDs

6/35


Concepts that must be addressed in approachinga solution to such problems:

distribution of residence times occurs

quality of mixing varies with position in reactor

a model must used to describe the phenomenon

Accounting for nonideality requires

knowledge of macromixing (RTD)

application of the RTD to a reactor (micromixing) to

predict reactor performance.

8/3/2019 Ch E 441 RTDs

7/35

RTD Functions

In any reactor, the RTD can affect performance

Ideal Plug Flow and Batch Reactors

Every atom leaving reactor is assumed to have resided inthe reactor for exactly the same duration. No axial mixing.

Ideal CSTR

Some atoms leave almost immediately, others remainalmost forever. Many leave after spending a period of timenear the mean residence time. Perfect mixing.

RTD is characteristic of mixing in a reactor.

RTDs are not unique to reactor type.Different reactor types can have the same RTD.

8/3/2019 Ch E 441 RTDs

8/35

Measurement of RTD

RTD is measured experimentally by use of aninert tracer injected into the reactor at t = 0.Tracer concentration is measured at effluent as afunction of time.

Tracer must be non-reactive and non-absorbingon reactor walls/internals.

Tracer is typically colored or radioactive to allow

detection and quantification. Common methods of injection are pulse and step

inputs.

8/3/2019 Ch E 441 RTDs

9/35

Pulse Input RTD Measurement

An amount of tracer No is suddenly (all at once)injected into the feed of a reactor vessel with

flow at a steady state.

Outlet concentration is measured as a function of

time.

reactor

injection detection

feed effluent

8/3/2019 Ch E 441 RTDs

10/35


reactor

injection detection

feed effluent

pulse injection

C

t0 +-

pulse response

C

t0 +-

8/3/2019 Ch E 441 RTDs

11/35


Injection pulse in system of single-input andsingle-output, where only flow (no dispersion)carries tracer material across system boundaries.

The amount of tracer materialN leaving the

reactor between t and t+t for a volumetricflowrate of is

where t is sufficiently small that the

concentration of tracer C(t) is essentially constantover the time interval.

ttCN

8/3/2019 Ch E 441 RTDs

12/35


Dividing by total amount of tracer injected, Noyields the fraction of material that has a

residence time between t and t+t:

where E(t) represents the residence-timedistribution function.

ttEtN

tC

N

N

oo

0 dttCtC

tE

8/3/2019 Ch E 441 RTDs

13/35

8/3/2019 Ch E 441 RTDs

14/35

Step Input RTD Measurement

In general, the output concentration from avessel is related to the input function by the

convolution integral (Levenspiel):

where the inlet concentration takes the form of

either a perfect pulse input (Dirac delta function),imperfect pulse injection, or a step input.

dt'tE'ttCtCt

0inout

8/3/2019 Ch E 441 RTDs

15/35


t

0o

t

0inout dt'tECdt'tE'ttCtC

Considering a step input in tracer

concentration for a system of constant :

0tC

0t0tC

oo

constant can be broughtoutside the integral

8/3/2019 Ch E 441 RTDs

16/35

Divide by Co

F(t) fraction of molecules that have spent a time

t or less in reactor (Cumulative age)

Differentiate to obtain RTD function E(t)


tF'dt'tECC t

0stepo

out

stepo

out

CC

dtdtE

8/3/2019 Ch E 441 RTDs

17/35


Advantages Easier to carry out experimentally than pulse test

Total amount tracer in feed need not be known

Disadvantages Often difficult to maintain a constant tracer

concentration in feed.

differentiation of data, often leads to large error.

Requires large amount of tracer, which in some cases

can be expensive.

8/3/2019 Ch E 441 RTDs

18/35

RTD Characteristics

E(t) is sometimes called the exit-age distributionfunction.

If the age of an atom is regarded as the amount

of time it spends in the reactor, E(t) is the age

distribution of the effluent.

E(t) is the most often used distribution function

for reactor analysis.

8/3/2019 Ch E 441 RTDs

19/35

Fraction of exit stream that has resided in the reactor

for a period of time shorter than a given value of t:

Fraction of exit stream that has resided in the reactorfor a period of time longer than a given value of t:

Integral Relationships

tFdttEt

0

tF1dttEt

8/3/2019 Ch E 441 RTDs

20/35

Integral Relationships

8/3/2019 Ch E 441 RTDs

21/35

Mean Residence Time

0

0

0m dttEt

dttE

dttEt

t

The nominal holding time, , is equal to the

mean residence time, tm. The mean value of the time is the first

moment of the RTD function, E(t).

can be used to determine reactor volume

8/3/2019 Ch E 441 RTDs

22/35

1st moment mean residence time

2nd moment variance (extent of spread of

the RTD)

3rd moment skewness (extent RTD is skewed

relative to the mean)

Other Moments of the RTD

0

2

m

2

dttEt-t

0

3m

13 dttEt-ts 23

8/3/2019 Ch E 441 RTDs

23/35

Normalized RTD Function, E()

t

A normalized RTD is often used to allow

comparison of flow profiles inside reactors of

different sizes, where

tEE

etEE

e1

tE tfor an ideal CSTR

8/3/2019 Ch E 441 RTDs

24/35

Internal-Age Distribution, I()

Fraction of material inside the reactor that has

been inside for a period of time between and

+

0 dE1

1

I

8/3/2019 Ch E 441 RTDs

25/35

RTD in a Batch or PFR

Simplest case

Spike at t = (or = 1) of infinite height and zero

width with an area of one

ttE

0x0x0x

1dxx

gdxxxg

8/3/2019 Ch E 441 RTDs

26/35

Effluent concentration is identical to that of

reactor contents.

A material balance for t > 0 on inert tracer

injected as a pulse at t = 0

RTD in a CSTR

dt

dC

VC0

accout-in

t

0eCtC

8/3/2019 Ch E 441 RTDs

27/35

RTD in a CSTR

Recall definition of E(t), and substitute:

dt

dC

VC0

t

0eCtC

t

0

t

0

t0

0

e

dteC

eC

dttC

tCtE

8/3/2019 Ch E 441 RTDs

28/35

Ideal Reactor Response to Pulse

E

t

Batch/PFR

E

CSTR

1

1

8/3/2019 Ch E 441 RTDs

29/35

Laminar Flow RTD

2

2

o

2

maxR

r1

R

2

R

r1UU

Velocity profile in a pipe (cylindrical

coordinates) is parabolic according to:

Time for passage of an element of fluid is

22

o

2

Rr11

2Rr11

2LR

rULrt

8/3/2019 Ch E 441 RTDs

30/35

The fraction of total fluid passing between r

and r+dr is d/0:

Laminar Flow RTD

00

rdr2rUd

rdr

R

t4rdr

Rr1

2

R

4dt

2

22

22

2

Rr11

2rt

8/3/2019 Ch E 441 RTDs

31/35

Laminar Flow RTD

Combining

dtt2

dtt4R2

tLrdr2

tLd

3

2

2

2

000

00

rdr2rUd

rdr

R

t4rdr

Rr1

2

R

4dt

2

22

22

8/3/2019 Ch E 441 RTDs

32/35

Laminar Flow RTD

The minimum time the fluid will spend in the

reactor is

Therefore, the complete RTD function is

22

V

R

R

U2

L

U

Lt

02

2

avgmax

23

22

tt2

t0

tE

5.021

5.00

E 3

8/3/2019 Ch E 441 RTDs

33/35

Laminar Flow RTD

The RTD appears graphically as

5.02

15.00

E

3E

1

0.5

8/3/2019 Ch E 441 RTDs

34/35

RTD of PFR and CSTR in series

CSTR (s) followed by PFR (p)

CSTR output will be delayed by a time ofp

p

s

t

p

te

t0

tE sp

8/3/2019 Ch E 441 RTDs

35/35

RTD of PFR and CSTR in series

PFR (p) followed by CSTR (s)

PFR output will delayed the introduction of the pulse

to the CSTR by a time ofp

Regardless of the order, the RTD is the same. However, theRTD is not a complete description of structure for a particular

reactor or system of reactors (see Example 13-4).

p

s

t

p

te

t0

tE sp

Ch E 441 RTDs

Documents