Post on 24-Oct-2014
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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