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Centre for Catalysis Research

Department of Chemical EngineeringUniversity of Cape Town • Rondebosch 7701

South Africa

Residence time distribution in ideal reactors

Linda H. CallananEric van Steen

1. Residence time2. When to determine RTD?3. How to determine RTD?4. Evaluation of RTD5. RTD of PFR/CSTR


Residence Time• How does long each molecule stay in the reactor (the residence

time)?

We need to know enough about the fluid to design the reactor. In most cases it is sufficient to know the average residence time

• Residence time distribution function (E-function) tells us quantitatively how long each fluid element spends in the reactor.– E(t) probability of residence time τ=t in unit– E.dt the fraction of fluid with an age between t and t + dt

in the exit stream.

vV

=τreactor through rateflow volumetric

volume actorRe=τ


Residence Time (2)

Time, s

E(t),

s-1

t

Probability that fluidspends time t in unit1dtE

0=⋅∫

∞


When to determine RTD?

• Laboratory reactors:typically assumed to be idealsometimes deviations determined experimentally

• Pilot-scale reactorsTake into account non-ideality of real reactors in scale-up

• Demonstration unitsTake into account non-ideality of real reactors in scale-up

• Production unitsTest to investigate observed non-design behaviour of unit(s)


When to determine RTD?

• Basic reactor design Assume ideal flow behaviour (complete back mixing CSTRno back mixing PFR)

• Deviation from the ideal can be considerable.

• Causes of non-ideal flow:– Channelling – Recycling– Bypassing– Stagnant zones

O. Levenspiel, “Chemical Reaction Engineering,” 2nd Ed., John Wiley & Sons, 1972, pp 254


How to determine RTD?

• The RTD is determined by injecting a tracer into the feed stream of a reactor and measuring its concentration in the outlet stream.

• The tracer can take various forms, but must have certain properties:– Inert (Not allowed to react with other components in the reactor feed)– Easily measurable

• Examples of tracers include– Chemical molecules or atoms– Radioactive molecules– Energy (temperature)


Measuring the RTD

•The are two methods for determining the residence time distribution :

•Pulse inputA finite amount of an inert tracer is injected at t=0 into a feed stream in a short period of time. The outlet concentration is measured as a function of time.

•Step inputThe concentration of the tracer is increased at t=0. The change in the concentration in the outlet is measured as a function of time


Measuring the RTD – Pulse input

Injection of n molesof tracer

Unitv

time

C

( )vndttC0 =⋅∫

∞

( )( )

( )

vn

tC

dttC

tC)t(E0

==∫∞

time

C


Measuring the RTD - step input

A constant amount of tracer is added to the feed stream.

–The F curve represents the fraction of fluid exiting the reactor that is younger than age t1:

( )⎩⎨⎧

≥<

=0t)ttancons(C0t0

tCo

0

( )0CtC)t(F =

step0C)t(C

dtd

dtdF)t(E ⎥

⎦

⎤⎢⎣

⎡==E curve is the differential of the F curve

The F curve0

0.5

1

0 10 20 30 40Time

F


Measuring the RTD

• Pulse input:– All of the tracer injected must be accounted for

• Step Input: – The concentration finally reached must equal the inlet

concentration

( ) ( )tCtC 0=∞

( ) aliquotaliquot,tracer0 vCdttC ⋅=∫∞

With either method care must be taken that the mass balance closes around the unit


RTD from experimental dataPulse input

The following data represent a continuous response to a pulse input into a closed vessel that will be used as a chemical reactor. Plot the exit age distribution E.

Time t, min 0 5 10 15 20 25 30 35

Tracer Output concentration, g/l 0 3 5 5 4 2 1 0


Use trapezoidal rule to improve accuracy

Time t, min

Ctracer g/l Cavg ∆t Cavg * ∆t E Eavg Eavg * ∆t

0 0 1.5 5 7.5 0 0.015 0.0755 3 4 5 20 0.03 0.04 0.2

10 5 5 5 25 0.05 0.05 0.2515 5 4.5 5 22.5 0.05 0.045 0.22520 4 3 5 15 0.04 0.03 0.1525 2 1.5 5 7.5 0.02 0.015 0.07530 1 0.5 5 2.5 0.01 0.005 0.02535 0 0

Q: 100 Sum: 1


Check results by integrating E*dt



0

0.02

0.04

0.06

0 10 20 30 40

Time, min

E(t)


RTD from experimental datastep input

Berty reactor


RTD from experimental datastep input

0

0.2

0.4

0.6

0.8

1

0 1000 2000 3000

Time, s

F(t)

= C

(t)/C

0

0

1

2

3

0 1000 2000 3000

Time, s

y CO

2, vo

l-%

C0

0

0.001

0.002

0.003

0 1000 2000 3000

Time, s

E(t)

= dF

(t)/d

t


Evaluation of RTD

∫∫∫ ∞∞

∞

⋅⋅=⋅

⋅⋅= 0

0

0m dtEt

dtC

dtCtt

( )∫∞

⋅⋅−=σ 02

m2 dtEtt

Mean Residence timeThe mean or average residence time:It can be derived that the mean residence time is simply equal to the space time:

VarianceIt is often more interesting to know how much of a spread in theresidence time distribution there is. This is commonly measured by the variance:

SkewnessThis is not often used but measures the extent to which the distribution is skewed to one side or another.

vVtm =τ= But be careful with dead zones

( )∫∞

⋅⋅−⋅σ

= 03

m2/33 dtEtt1s


RTD in Ideal ReactorsPlug flow reactors (PFR)

For perfect plug flow, all molecules spend exactly the same amount of time in the reactor (simplest type of reactor to consider).

The residence time distribution follows a Dirac delta function:

( ) ( )

( )⎩⎨⎧

=∞≠

=δ

τ−δ=

0xwhen0xwhen0

x

ttE

O. Levenspiel, “Chemical Reaction Engineering,” 2nd Ed., John Wiley & Sons, 1972


RTD in Ideal ReactorsContinued stirred tank reactor (CSTR)

In ideal CSTR, concentration at all points in the reactor identical

The unsteady state mass balance over the reactor, after an injection at t=0 is then:

in – out = accumulation

–C is both the reactor and outlet concentration.•Integrating with C = C0 at t=0 gives the concentration time profile:

dtdCVCv0 ⋅=⋅−

τ−⋅= /t0 eC)t(C

( )( ) τ

=⋅

⋅==

τ−

∞ τ−

τ−

∞∫∫

/t

0/t

0

/t0

0

e

dteC

eC

dttC

tC)t(E


RTD in Ideal ReactorsContinued stirred tank reactor (CSTR)

Characteristics of E-curve:Area 1Intercept: 1/τ

O. Levenspiel, “Chemical Reaction Engineering,” 2nd Ed., John Wiley & Sons, 1972,


RTD in Ideal ReactorsTank in series model

C0v0

C1v1

C2v2

C3v3

V1 V2 V3

A reactor can often be considered as a series of perfectly mixedzones (tanks) with the fluid moving from one zone to the next.

Typically, the zones are assumed to be equally sized

PFR can be viewed as an infinite number of CSTRs in series


RTD in Ideal ReactorsTank in series model

Considering outlet from the first tank:

For the second tank:

This can be integrated to give

Similarly for the third tank:

ii V/vt0

/t01 eCeC)t(C ⋅−τ− ⋅=⋅=

τ=

τ− /te)t(E

Normal CSTR

2/t

0212

i CveCvCvCvdt

dCV i ⋅−⋅⋅=⋅−⋅= τ−

Mole Balance

i/t

i

02 etC)t(C τ−⋅

τ⋅

=

i/t2i

20

3 e2

tC)t(C τ−⋅τ

⋅=


RTD in Ideal ReactorsTank in series model – E-curve

( ) i/t3i

2

3

3 e2t

dt)t(C)t(CtE τ−⋅

τ==

∫For 3 reactors in series

( )( )

i/t

in

1ne

!1nttE τ−−

⋅τ−

=In general

0

0.5

1

1.5

0 0.5 1 1.5 2 2.5 3

Time

E(t)

n=1

n=2

3 5 10100 (*0.25)


RTD in Ideal ReactorsRecycle reactor

E θ(o

r Cθ)

Rυ

(R+1)υυ V

b 2b 3b 4b

E θ(o

r Cθ)

Rυ

(R+1)υυ V

b 2b 3b 4b


Parabolic flow profile: fluid in the centre spends the shortest time in the reactor.

The velocity profile is given as follows:

Umax is the centre velocityUavg is the average velocity through the tube

The time spent in the reactor for the fluid element at radius r is:

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−

π=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−=

2

2

2

avg

2

max Rr1

Rv2

Rr1U2

Rr1UU

( )[ ] ( )[ ]220

2

R/r12R/r1L

2R

)r(UL)r(t

−

τ=

−⋅

υπ

==2U

Ltmax

minτ

==

RTD in Ideal ReactorsLaminar flow reactors


RTD in Ideal ReactorsLaminar flow reactors

The fraction of fluid moving through the shell between r and (r + ∆r) is:

E-curve for laminar flow (see e.g. Fogler 13.4.3)

00 vdrr2)r(U

vdv π

=

⎪⎪⎩

⎪⎪⎨

⎧

τ≥

τ

τ<

=

2t

t2

2t0

)t(E

3

2E

(t)

τ2


Segregation Model Conversion from tracer information

Valid for linear processes: Each fluid element does not react with any other

Mixing occurs as late as possible (at the reactor exit)A number of flow patterns may give the same RTD curve.This is not a problem with linear processes, as all will give the same conversion.

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

+

⋅

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

+

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

∑

dttandtbetween

ageofstreamexitoffraction

dttandtbetweenageof

elementinleftttanreacof.conc

streamexitinionconcentrat

mean

elementsexitAll

∫∞=

=0t element,AA dtECC [H. Scott Fogler, “Elements of

Chemical reaction Engineering” 4th

Ed., Prentice Hall, 2006]


Types of Reactions

Irreversible first order reaction:

Reversible first order reaction:

Successive first order reactions: A R P

These equations can then be solved if E is known.

tk0,Aelement,A

0,A

element,A eCCortkC

Cln ⋅−=⋅−=

∫∞=

⋅−= 0ttk

0,AA dtEeCC

[ ]∫∞=

⋅−+−= 0tX/tk

AeAe0,AA dtEeXX1CC Ae

[ ]∫∞=

⋅−⋅− −−⋅

= 0ttktk

12

10,AR dtEee

kkkC

C 21

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