residence time distribution in real reactors - uni oldenburg · residence time distribution in real...

Centre for Catalysis Research

Department of Chemical EngineeringUniversity of Cape Town • Rondebosch 7701

South Africa

Residence time distribution in real reactors

Linda H. CallananEric van Steen

1. Recap on RTD and conversion in real reactors

2. Dispersion model3. Tank-in-series model4. Compartment model


Residence Time Distribution (RTD)

Determined by introducing a non-reactive, non-adsorbing tracer into the feed and measuring the concentration of the tracer in the exit line.

Pulse input

Step input

( ) ( )

( )∫∞=

=

⋅

= t

0tdttC

tCtE

( ) ( )( )dt

tFdtE =( ) ( )( )0tCtCtF

inlet >=

Cex

it(t)

t

E(t)

t

Cex

it(t)

t

F(t)

t

E(t)

t


Evaluation of Residence Time Distributions (RTD)

Consistence checkPulse input

Step input

Mean Residence timeThe mean or average residence time:The mean residence time should be equal to the space time:

VarianceSpread in the residence time distribution is commonly measured by the variance:

vVt

?m =τ=

∫∞=

=

⋅⋅=t

0tm dtEtt

fluid

injectedt

0t vn

dtC =⋅∫∞=

=

( ) inletCtC =∞=

( )∫∞

⋅⋅−=σ 02

m2 dtEtt


Conversion in real reactors -Segregation Model

The fluid consisting of non-interacting elements

Each exit stream is thought to consist of elements having spent various times in the reactor.

The exit concentration of the reactant is given by:

Assumptions:Valid for linear processes: Each fluid element does not react with any other Mixing occurs as late as possible (at the reactor exit)

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

+

⋅

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

+

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

∑

dttandtbetween

ageofstreamexitoffraction

dttandtbetweenageof

elementinleftttanreacof.conc

streamexitinionconcentrat

mean

elementsexitAll

[H. Scott Fogler, “Elements of Chemical reaction Engineering” 4th

Ed., Prentice Hall, 2006]

∫∞=

=

⋅⋅=t

0telement,AA dtECC


Determination of conversion from E-curve

The residence time distribution in a reactor has been determinedand the E(t) data have been determined as shown below:

The reactor is to be used for a liquid phase decomposition. The rate of reaction is 1st order with respect to the reactant (k = 0.1 min-1). Determine:

Time t, min 0 5 10 15 20 25 30 35E, min-1 0 0.03 0.05 0.05 0.04 0.02 0.01 0

1. The mean residence time2. The conversion, which would have been obtained if the reactor

is an ideal PFR/CSTR3. The conversion obtained in this reactor according the

segregation model


Determination of mean residence time from E-curve

∫∞=

=

⋅⋅=t

0tm dtEttMean residence time:

t, min 0 5 10 15 20 25 30 35

E, min-1 0 0.03 0.05 0.05 0.04 0.02 0.01 0

t.E 0 0.15 0.50 0.75 0.80 0.50 0.30 0

Integral 0.375 1.625 3.125 3.875 3.25 2.00 0.75

( ) ( )( ) ( )1212 tttEtE21

−⋅⋅+⋅⋅

Mean residence time: 15 min


Determination of conversion in ideal reactors

PFR:

CSTR

0,A

A

0,A

ACvr

Fr

dVdX

⋅−

=−

=0,A

AC

rddX −

=τ

1st order reaction: ( )X1kC

CkC

rddX

0,A

A

0,A

A −⋅=⋅

=−

=τ

( ) τ⋅=−− kX1ln τ⋅−−= ke1XConversion in ideal PFR: 77.8 %

0,A

A

0,A

ACvr

Fr

VX

⋅−

=−

=0,A

AC

rX −=

τ

1st order reaction: ( )X1kC

CkC

rX0,A

A

0,A

A −⋅=⋅

=−

=τ τ⋅+

τ⋅=

k1kX

Conversion in ideal CSTR: 60 %


Conversion using segregation model

Segregation model

The concentration of the reactant in each element depends on thetime it spent in the reactor. Each element can be seen as a batch-type reactor.

The average concentration of the reactant in the exit stream is given by:

tk0,Aelement,A eCC ⋅−⋅=

∫∞=

=

⋅⋅=t

0telement,AA dtECC

∫∞=

=

⋅− ⋅⋅⋅=t

0t

tk0,AA dtEeCC

∫∞=

=

⋅− ⋅⋅−=−=t

0t

tk

0,A

A dtEe1CC1X


Conversion using segregation model

t, min 0 5 10 15 20 25 30 35

E, min-1 0 0.03 0.05 0.05 0.04 0.02 0.01 0

e-k.t.E 0 0.018 0.018 0.011 0.005 0.002 0.000 0

Integral 0.045 0.091 0.074 0.041 0.018 0.005 0.001

( ) ( )( ) ( )121tk

2tk ttEeEe

21 12 −⋅⋅+⋅⋅ ⋅−⋅−

X = 1-Σintegral

Comparison of conversions:PFR 77.8%CSTR 60.0%Segregation model 72.4%


Dispersion model

convectionTc

dispersion

TcT CAu

dzdCADF ⋅⋅+⋅⋅−=

Pulse input

t

C(t)

t

C(t)

moleculardiffusionfluid flow

t

C(t)

t

Flux of inert tracer:

Mole balance on inert tracer:

FT: molar flux of tracerD: diffusion coefficient for tracerAc: cross sectional area of tubez: lengthu: linear velocityt

CAz

F Tc

T∂∂⋅=

∂∂

−


Development of dispersion model

tC

zCu

zCD TT2T

2

∂∂

=∂∂⋅−

∂

∂⋅Mole balance on inert tracer:

θ⋅= duLdtL

utt ⋅=τ=θIntroducing a dimensionless time:

a dimensionless length: Lz=λ λ⋅= dLdz

2222 dLdL2dzz2dz λ⋅=λ⋅λ⋅⋅=⋅⋅=

Substituting dimensionless numbers in mole balance on inert tracer:

θ∂∂

=λ∂

∂−

λ∂

∂⋅

⋅TT

2T

2 CCCLu

D



DLu ⋅

=Bo

LD

θ∂∂

=λ∂

∂−

λ∂

∂⋅

⋅TT

2T

2 CCCLu

D

Bodenstein number Bo (also called Per)

Ratio of rate of convective transport relative to rate of transport by diffusion

Convective transport large (Bo ∞) PFR-behaviourTransport by diffusion ((Bo 0) mixing by diffusion CSTR-behaviour

For constant conditions (temperature/pressure, etc.), Bo increases with increasing L (length of reactor)

Long reactors, B0 ∞ approaching plug flow behaviour!

moleculardiffusionfluid flow

Characteristic rate of diffusion:

Characteristic rate of convective flow: u



θ∂∂

=λ∂

∂−

λ∂

∂⋅ TT

2T

2 CCC1Bo

(discontinuity in Bo at λ=0 and λ=1)Closed system

λ=0 λ=1

(discontinuity in Bo at λ=0)Closed-open system

Open system

τ=t ( )⎟⎠⎞

⎜⎝⎛ −⋅−⋅=σ −Bo

2BoBoe122t22

( )Bo+⋅τ= 1t ⎟⎠⎞

⎜⎝⎛ +⋅=σ 3BoBo

32t22

τ=t ⎟⎠⎞

⎜⎝⎛ +⋅=σ 2BoBo

82t22

( )θ⋅⋅θ−−

⋅θ⋅π

⋅= 41 2

e21E

BoBo


Dispersion modelcomparison open and closed systems

0

0.2

0.4

0.6

0.8

1

0.0001 0.01 1 100 10000

Bodenstein number

Rel

ativ

e va

rianc

e,

σ2 cl

osed

/ σ2 op

en

DLu ⋅

=Bo

Insignificant change forBo >50

For Bo>50 all systems can beconsidered to be open systems(i.e. solvable!)


Dispersion modelopen systems

0

2

4

6

8

0 0.5 1 1.5 2

Relative time, θ=t.u/L

E(θ)

Bo = 0.01 ~ 0(mixed flow; enlargement *10)

Bo = 5

Bo = 50

Bo = 500

Bo = 50000 ~ ∞(plug flow)


Determination of Bo from E-curve

⎟⎠⎞

⎜⎝⎛ +⋅=σ 2BoBo

82t22

( ) 20

2m

2 min5.47dtEtt ⋅=⋅⋅−=σ ∫∞


Time t, min 0 5 10 15 20 25 30 35E, min-1 0 0.03 0.05 0.05 0.04 0.02 0.01 0

Assuming an open system: Bo = 12.5

Average residence time: 15 min

Variance:

Assuming a closed system: Bo = 8.3( )⎟⎠⎞

⎜⎝⎛ −⋅−⋅=σ −Bo

2BoBoe122t22


Conversion and dispersion model (1st order rxn – Closed system)

Damköhler number:

1st order reactions:

Solving the mole balance of species A in the reactor (closed system) (second order differential equation) :

τ⋅⋅= −1n0ACkDa

( ) ( ) 2q

22q

2

2

0A

L,A

eq1eq1

eq4CC

X1 ⋅⋅

⋅−−⋅+

⋅⋅==− Bo-Bo

Bo

BoDa41q ⋅

+=

τ⋅= kDa


Conversion and dispersion model (1st order rxn – Closed system)

0%

20%

40%

60%

80%

100%

0.001 0.01 0.1 1 10 100 1000

Bodenstein number, Bo

Con

vers

ion

for 1

st o

rder

re

actio

n

Da = 10Da = 5

Da = 2

Da = 1

Da = 0.5

Da = 0.2Da = 0.1

The reactor (average residence time: 15 min) is to be used for aliquid phase decomposition. The rate of reaction is 1st order with respect to the reactant (k = 0.1 min-1). Da = 1.5 (Boclosed system = 8.3)

Comparison of conversions:PFR 77.8%CSTR 60.0%Segregation model 72.4%Dispersion model 73.2%


Conversion and Dispersion model (1st order rxn)


Tank-in-series model

C0v0

C1v1

C2v2

C3v3

V1 V2 V3

Reactor is assumed to contain n equally sized CSTRs in series (PFR can be viewed as an infinite number of CSTRs in series)

( )( )

i/t

in

1ne

!1nttE τ−−

⋅τ−

=n

22 τ=σ


Determination of number of CSTRs in series from E-curve

( ) 20

2m

2 min5.47dtEtt ⋅=⋅⋅−=σ ∫∞


Time t, min 0 5 10 15 20 25 30 35E, min-1 0 0.03 0.05 0.05 0.04 0.02 0.01 0

Assuming tank in series model:n = 4.7

Average residence time: 15 min

Variance:

n

22 τ=σ


Conversion from tank-in-series model

i

i1ii Ck

CC⋅−

=τ −

( )i

1ii0i r

XXC−−⋅

=τ −

i1i

ik11

CC

τ⋅+=

−

Performance equation CSTR:

For 1st order reaction:

Assuming tank in series model (n = 4.7; τi = 15/4.7 = 3.16 min; k.τi = 0.3) X= 72.8%

( ) nni0

i

nk1

1k11

CCX1

⎟⎠⎞

⎜⎝⎛ τ

⋅+

=τ⋅+

==−

Comparison of conversions:PFR 77.8%CSTR 60.0%Segregation model 72.4%Dispersion model 73.2%Tank-in-series model 72.8%


Compartment models

Considering the actual reactor as a set of ideal set-ups:1. Ideal reactor(s) with dead volume2. Ideal reactor(s) with by-pass

4. Ideal reactors in series (tank-in-series model) 5. Combinations of dead volume, bypassing, ideal reactors in

series/parallel

3. Ideal reactor(s) in parallel


Compartment modelsIdeal reactors with dead volume

Vplug

Vdead

Vreactor = Vplug + Vdead

VCSTR

Vdead Vreactor = VCSTR + Vdead

v v

vVt reactorectedexp =

( )t

Vv

CSTR

CSTReV

vtE⋅−

⋅=

E(t)

E(t)

vV

t plug=


Compartment modelsIdeal reactors with bypassing

Vplug

v = v1 + v2

VCSTR

v

v

v1

v2v2

v1

E(t)

vvArea 1=

vvArea 2=

( )t

Vv

CSTR

1 CSTR1

eV

vtE⋅−

⋅=

1

plugv

Vt =

vvArea 1=E

(t) vvArea 2=


Compartment modelsideal PFRs in parallel

Vplug,1v v1

v2Vplug,2

v = v1 + v2

1

1,plug

vV

t =

vvArea 1=E

(t)

vvArea 2=

2

2,plugv

Vt =


Compartment modelsideal CSTRs in parallel (1)

VCSTR,1E

(t)

v1

( )t

Vv

2,CSTR

2t

Vv

1,CSTR

1 2,CSTR2

1,CSTR1

eV

veV

vtE⋅−⋅−

⋅+⋅=

VCSTR,2

v2


Compartment modelsideal CSTRs in parallel (2)

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5time

E(t)

V1 = 0.75 * Vtotalv2 = 0.1 * v

VCSTR,1

v

VCSTR,2

v2

0.01

0.1

1

0 1 2 3 4 5time

E(t)

V1 = 0.75 * Vtotalv2 = 0.1 * v


Compartment modelscombination of models

Vd

Vp

Eθ

(or C

θ)

Vm

υ2

υ1υ

VVP

1υυ

υυ1area =

υυ2area =

Vd

Vp

Eθ

(or C

θ)

Vm

υ2

υ1υ

VVP

1υυ

υυ1area =

υυ2area =


What is the “best” model

Dispersion model:Gives insight in the design phase to anticipate non-ideal behaviour and the consequences

Compartment model“Visualizes” the origin of possible non-ideal behaviour:

by-passingdead volumemixing zones

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