Basics of Non-Ideal Flow

Duvvuri Subbarao

Importance of flow structure in Process Equipment

In all types of process equipment, such as heat exchangers, packed columns, and reactors, the type of flow affects the performance of the unit.

Deviation from the two ideal flow patterns, PFR and CSTR, can be causedby channeling of fluid,by recycling of fluid, or by creation of stagnant regions in the vessel..

Importance of Velocity Profiles• In an ideal PFR, it is assumed that all the molecules

flow at a uniform velocity (flat velocity profile) and have the same residence time

• In a tubular reactor, velocity at the wall has to be zero and center velocity is more than the average velocity and so residence time can not be same for all the molecules.

• Conversion depends on the residence time.• Residence time distribution (RTD) depends on velocity

profiles.• Computing actual velocity profiles under the reaction

conditions in industrial reactors is impractical, even in today's computer age.

Stimulus-Response Technique

• The non ideal flow in industrial flow vessels can be determined easily and directly by a widely used method of inquiry, the stimulus-response experiment.

• To introduce this topic, we will only consider steady-state flow, without reaction and without

density change, of a single fluid through a vessel with single inflow

and one single outflow. (Danckwerts, 1953),

Minimum Information needed• In a plug flow reactor, all the molecules leaving the reactor spend

exactly the same time with no age distribution.

• Suppose in an industrial reactor, f1 fraction of the flowing fluid through the reactor spends some time 1, another f2 fraction spends time 2. Conversion for each fraction can be calculated considering that each stream to be in plug flow with their respective residence times.

• Information on residence time for each fraction of the flowing fluid is required; this information is called the, Residence time distribution.

keX1

21 k

2

k

1 efefX1

Experimental Methods (Nonchemical) for Finding Exit Age Distribution

(Residence time distribution)

Pulse response for ideal Reactors

C

F

Out

In

CSTR

1 2

In

Time

PFR

Out

In

Time

Non-ideal Reactor

Pulse Technique & C pulse curve

Consider a vessel of volume V m3

through which flows F m3/s of fluid.

Instantaneously introduce M units of tracer (kg or moles) into the fluid entering the vessel,

and record the concentration-time of tracer leaving the vessel.

Developing E – Curve

F is same as vT is same as M

=T/F

Equations for Estimating Exit Age Distribution

0

0 0

1

tracer

tracer

tracer

tracer

Kg of tracer in the exit stream F C

Cumulative total tracer in the exit stream T F C dt

Cdt E dt

TF

CE

TF

Mean Residence timeFrom the material balance for the vessel we find

F is same as vT is same as M

T/F

V/F

E Curve to E Curve

Definition of E, THE AGE DISTRIBUTION OF FLUID / THE RTD

the fraction younger than age t, is

fraction of material older than t,,

E.dt is the fraction of material collected over a period of dt

Then all the material together is

EXAMPLE 1 E curve, Average Residence Time

Response to a pulse input of tracer to a flow reactor is given in the table. Calculate Mean residence time of the fluid in the reactor

0 0 - 0 0 05 3 5 15 75 0.0310 5 5 25 250 0.0515 5 5 25 375 0.0520 4 5 20 400 0.0425 2 5 10 250 0.0230 1 5 5 150 0.0135 0 5 0 0 0

E.Δt=1.0

min

tttracerC

3

.

.min

C t

Kg

m

t C t

100T

F

C t

1500tC t 1500

15 min100

t C tt

C t

TF

CE tracer

CONVERSION IN NON-IDEAL FLOW REACTORS

• To evaluate reactor behavior in general we have to know four factors:

• 1. the kinetics of the reaction• 2. the RTD of fluid in the reactor

Conversions for PFR/Batch reactors

EXAMPLE 1 Conversion in a Non ideal reactor

Estimate Conversion in a CSTR, PFR and a reactor having non-ideal flow (C curve data based on pulse response given in the next slides)For a 1st order reaction Data: Reaction rate 1

, 0.307 , 15 minminA r A rr k C k

1

1 1

11

1 10.1785

1 0.307 15 5.6

82.15%

A in A A

A in A

A

A in

A

A in

F C F C V K C

VC C K

F

CVC KKF

C

C

Conversion

For a CSTR

0.307 15

ln

0.01

99%

AA

AA

A

A

A

A in

VK KA F

A in

dCK C

dtdC

F K CdV

dC dVK

C F

VC KC F

Ce e e

C

Conversion

For a PFR

EXAMPLE 1 continued Conversion in a Non ideal reactor

(Kr) t Exp[-kr t]Conversion based on PFR

E Δ t EXP((-kr) t)

0 0 0 -5 1.535 0.2154 0.03 5 0.032310 3.07 0.0464 0.05 5 0.011615 4.605 0.01 0.05 5 0.002520 6.14 0.0021 0.04 5 0.000425 7.675 0.0005 0.02 5 0.000130 9.21 0.0001 0.01 5 035 0 0 0 5 0

Sigma Edt =1.0

CA/Ca0=0.0469

tracer

E

CTF

min

tt

95.31%Conversion

Step change – F Curve

F

F

IN

OUT

Time

1 2

PFR

Time Time

Industrial Reactor

The Step ExperimentThe dimensionless form of the Cstep curve is called the F curve.

F-Curve

Relationship between the F and E Curves

But the first term is simply the F value, while the second is given by Eq. 1. So we have, at time t,

at time t = 0 switch to red and record the rising concentration of red fluid in the exit stream - the Fcurve.

At any time t > 0 red fluid and only red fluid in the exit stream is younger than age t. Thus we have

Conversion of F to E Curve

EXAMPLE 1 E curve, to F curve

Response to a pulse input of tracer to a flow reactor is given in the table. Calculate Mean residence time of the fluid in the reactor

0 0 - 0 0 0 05 3 5 15 0.03 0.15 0.1510 5 5 25 0.05 0.25 0.415 5 5 25 0.05 0.25 0.6520 4 5 20 0.04 0.2 0.8525 2 5 10 0.02 0.1 0.9530 1 5 5 0.01 0.05 1.035 0 5 0 0 0 0

1.0

min

t

TF

tracer.C

tracer.C

tC

CEt tC tracer.C tracer.CC tE t.EF

100tCT

Ftracer

EXAMPLE 1 F curve, to E curve

Response to a pulse input of tracer to a flow reactor is given in the table. Calculate Mean residence time of the fluid in the reactor

0 0 0 - -

5 3 0.15 0.15 5 0.03

10 8 0.4 0.25 5 0.05

15 13 0.65 0.25 5 0.05

20 17 0.85 0.2 5 0.04

25 19 0.95 0.1 5 0.02

30 20 1.0 0.05 5 0.01

35 20 1 0 5 0

min

tt

FE

ttracer.FC

max.tracer.F

tracer.F

C

CF F

Summary

Compartment ModelF

F

1

1

1

i

i

i

F C F C V K C

VC C K

F

CVC KF

1 2

0

1

0

. .

.

0,

0, '

at

i

Lk

uexit

in

F C F C X A k C

F C CkC

A X

as X

dCu kC

dX

Ce

C

as X number of CSTR s in series

0

0

31 2

1 2 1

0 0

( )

.1

1

1

....

1 1

1 1

i

i

n n

in in n

Ln

x

exit

in

For each differential element compartment

A X XC C KC C K

F u

C

C XK

u

for n number of compartments

C C CC C

C C C C C

CX XC K Ku u

1

1

1

i

i

i

F C F C V K C

VC C K

F

CVC KF

Residence Time for Laminar Flow in Pipes

2 2

max

2

2

min

1 2 1

2 1

1

2 1

0, 2 , 0.5

, 0,

For La ar Flow

r ru u u

R R

L Lt

u ru

R

V L

F ut u

u rR

tr u u

tr R u

4

R

r2

R

r2

rR

0rd2Cu

rr

0rd2Cu

inCexit

C

Response to Step Change for Laminar Flow in a Circular Pipe

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5

t/t

C/C

0

Series1

Dispersion Model

1 2

Length

0 01 2

2

0 2

2

02

2

2

0

0

( )( ) 0

( ) 0 '

( ) 0

( ) 0

;

o

L L

L

nL

o

o

input rate output rate disappearance by reaction

u C N u C N X r

u C Nr

X

C C Cu D r Fick s Law N D

X X X

C CD u r

X X

D C Ck C

u L Z Z

X LZ

L u

General Model

CSTR Industrial Reactors PFR

1

L

oL

n

D

u

0L

oL

n

D

u

V=Volume of reactor

Plug Mix deadV V V V

By Pass

Recycle

Microfluids in plug or mixed flow

•plug flow and mixed flow.•one or the other often is optimum no matter what we are designing for.•these two patterns are simple to treat.

effect on overall behavior for a single flowing fluid

the earliness and lateness of mixing of material in the vessel.

Earliness of Mixing

• However, for a system with two entering reactant streams it can be very important.

Macro fluids

the state of aggregation of the flowing material, its tendency to clump andfor a group of molecules to move about together

Macrofluids - Fluid flowing as little clumps

• Each clump stays for different amount of time and reacts away as a little batch reactor ,

• Each clump will have different compositions depending on the time it spends in the reactor.

• So the mean composition in the exit stream will have to account for these two factors, the

• kinetics and the RTD. In words, then