Ch E 441 - Chemical Kinetics and Reaction Engineering

Residence Time Distributionsin Chemical Reactors

Residence Time Distributions• The assumption of a “perfectly mixed” reactor

often 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)

Residence Time Distributions• 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

Residence Time Distributions• 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 residence time for the PBR.

Residence Time Distributions• 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).

Residence Time Distributions• Concepts that must be addressed in approaching

a 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.

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 in

the reactor for exactly the same duration. No axial mixing.– Ideal CSTR

• Some atoms leave almost immediately, others remain almost forever. Many leave after spending a period of time near 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.

Measurement of RTD• RTD is measured experimentally by use of an

inert tracer injected into the reactor at t = 0. Tracer concentration is measured at effluent as a function of time.

• Tracer must be non-reactive and non-absorbing on reactor walls/internals.

• Tracer is typically colored or radioactive to allow detection and quantification.

• Common methods of injection are pulse and step inputs.

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

Pulse Input RTD Measurement

reactor

injection detection

feed effluent

pulse injection

C

t0 +-

pulse response

C

t0 +-

Pulse Input RTD Measurement• Injection pulse in system of single-input and

single-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 volumetric flowrate of is

where t is sufficiently small that the concentration of tracer C(t) is essentially constant over the time interval.

ttCN

Pulse Input RTD Measurement• Dividing by total amount of tracer injected, No

yields the fraction of material that has a residence time between t and t+t:

• where E(t) represents the residence-time distribution function.

ttEtN

tC

N

N

oo

0 dttC

tCtE

Step Input RTD Measurementstep injection

C

t

step response

C

t

reactor

injection detection

feed effluent

Step Input RTD Measurement• In general, the output concentration from a

vessel 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

Step Input RTD Measurement

t

0o

t

0inout dt'tE Cdt'tE 'ttCtC

• Considering a step input in tracer concentration for a system of constant :

0tC

0t0tC

oo

constant can be broughtoutside the integral

• 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)

Step Input RTD Measurement

tF'dt'tE C

C t

0stepo

out

stepo

out

C

C

dt

dtE

Step Input RTD Measurement• 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.

RTD Characteristics• E(t) is sometimes called the exit-age distribution

function. • 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.

• 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 reactor for a period of time longer than a given value of t:

Integral Relationships

tFdttE t

0

tF1dttE t

Integral Relationships

Mean Residence Time

0

0

0m dttE t

dttE

dttE tt

• 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

• 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

2m

2 dttEt-t

0

3m

13 dttEt-t s 23

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

Internal-Age Distribution, I()• Fraction of material inside the reactor that has

been inside for a period of time between and +

0dE1

1I

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

0x

0x0x

1dxx

gdxxxg

• 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

dCVC0

acc out -in

t0eCtC

RTD in a CSTR• Recall definition of E(t), and substitute:

dt

dCVC0

acc out -in

t0eCtC

t

0

t0

t0

0

e

dteC

eC

dttC

tCtE

Ideal Reactor Response to Pulse

E

t

Batch/PFR

E

CSTR

1

1

Laminar Flow RTD

2

2o

2

max R

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

22o

2

Rr1

1

2Rr1

1

2

LR

rU

Lrt

• 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

2Rr1

1

2rt

Laminar Flow RTD• Combining

dtt2

dtt4

R2

t

Lrdr2

t

Ld3

2

2

2

000

00

rdr2rUd

rdr

R

t4rdr

Rr1

2

R

4dt 2

22

22

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

t0tE

5.0

2

15.00

E3

Laminar Flow RTD• The RTD appears graphically as

5.0

2

15.00

E3E

1

0.5

RTD of PFR and CSTR in series• CSTR (s) followed by PFR (p)

– CSTR output will be delayed by a time of p

ps

tp

te

t0

tE sp

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 of p

Regardless of the order, the RTD is the same. However, the RTD is not a complete description of structure for a particular reactor or system of reactors (see Example 13-4).

ps

tp

te

t0

tE sp