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NON IDEAL REACTORS

or some useful tricks to model real reactors..

Today’s lecture

• Non ideal reactors – real reactors..

• Residence times and residence time distributions• Step response and pulse responses

• E(t) och F(t)

• Mixing models

Real reactors

The CSTR and the PFR are two ideal extremes of flow behaviour. A real reactor may have• Stagnant zones

• By-passing

• Dispersion

A complete analysis of the reactor will require a complete description of the flow, the kinetics and the heat transfer. However, it is often sufficient with a relatively minor model correction – especially for linear systems.

Residence times and distributions

Central concepts are:

• F(t)

• E(t)

• Mean residence time and variance

• Tanks-in-series model and dispersions model

F(t)

• The fraction of the outlet flow which has spent a time less than t in the reactor

• F(t) is directly obtained from a step-change experiment

0/)()( ctctF

F(t)

For the CSTR

For a PFR

)()( tHtF

Heavyside function

/1)( tetF GV /where

2

E(t)

• E(t)dt = the fraction of outflow which has remained in the reactor for a time between t and t+dt

• E(t) is determined from a pulse experiment

0

)(

)()(

tc

tctE

E(t)

For the ideal CSTR we have

For the PFR we have

Dirac’s delta function

/1

)( tetE

)()( ttE

GV /där

Useful relations

dt

dFtE )(

0

)( dtttEt

• NOTE: E(t) does not unambiguously determine the reactor type! Many different reactors (or combination of reactors) can result in the same E(t)

A combination of a CSTR and a PFR with equal volumes will e.g. give the same E(t) reagardless of the order the reactors are placed in.

One-parameter models

• E(t) may be used to determine parameters in simple one- parameter models

• The most common of these models are the tanks-in-series model and the dispersion model

• Also other combinations of ideal reactors in series may give a good fit to the E(t) distribution.

The dispersion model

rdz

dcu

dz

cdD i

iii

2

2

0

Axial dispersion

An analytical solution may be obtained for certain kinetic expressions, e.g. first order kinetics

3

Dispersion model

DuLDuL

DuL

AAeaea

aecc

2222

2

011

4

BAAkcr

)/(41 uLDka

where

Reactor size needed to reach a certain conversion for PFR and dispersion model

Salmi et al, Chemical Reaction Engineering and Reactor Technology, 2010

Tanks-in-series model Reactor size needed to reach a certain conversion for PFR and tanks-in-series

Salmi et al, Chemical Reaction Engineering and Reactor Technology, 2010

Variance

0

22 )( dttEtt

The expected value for the second momentum is the varianceof the distribution function

2

0

2 )(

tdttEt

Variance

The variance – determined from the experimentally determined E(t) distribution – can be used to estimate the number of tanks in a tanks-in-series model or D in a dispersion model

2

2

t

N2

2

2

t

uL

D Tanks-in series model Dispersion model

For low values of D

4

To determine the degree of conversion in a reactor we need to know

• E(t)

• Kinetics

• Micromixing (segregated flow or micromixed flow)

• Time of mixing

How does micromixing affect the reaction rate?

Segregated flow

Micro mixed flow

Reactor volume = 2V

Assume n-th order reactionnkcr

nnsegr kVckVcR 21

nn

micro

ccVk

V

VcVcVkR

2

22

2 2121

n

nn

micro

segr

cc

cc

R

R

22 21

21

n=1 R segr = Rmicro

n<1 Rsegr < Rmicro

n>1 Rsegr > Rmicro

Thus:

How does micromixing affect the reaction rate?

Segregated flow

Micro mixed flow

Reactor volume = 2V

So..

• In cases of segregated flow it is sufficient to know E(t) and the kinetic expression

• For first order reactions it is also sufficient to know E(t)

• In all other cases, a mixing model is needed in addition to E(t) to determine the conversion of reactants in the reactor