ERT 208/4 REACTION
ENGINEERING:
Distribution of Residence
Times for Reactors
(PART B)
By; Mrs Hafiza Binti Shukor
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)
MEASUREMENT OF RTD…
 2 most used methods of injection :
A) pulse input
B) step input
C
Pulse response
Pulse injection
C
Step injection
C
t
Step response
t
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)
Step Input / Step Tracer
Experiment …
Formulated more general relationship between a time
varying tracer injection and the corresponding conc in the
effluent.
We shall state without development that the output conc
from a vessel is related to the input conc by the convolution
integral;
Cout (t )   Cin (t  t ' ) E (t ' )dt
t
0
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)
Step Input / Step Tracer
Experiment …
Analyze a step input in the tracer conc for a system with a
constant volumetric flowrate.
Consider a constant rate of tracer addition to a feed that is
initiated at time t=0. Before this time, no tracer was added to
the feed. Symbolically, we have
Co (t )
0
t<0
(Co) constant t>0
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)
Step Input / Step Tracer
Experiment …
The conc of tracer in the feed to the reactor is kept at this
level until the conc in the effluent is indistinguishable from
that in the feed; the test may then be discontinued.
Cin
Cout
Step injection
t
Step response
t
Because the inlet conc is a constant with time, Co we can
take it outside the integral sign, that is
C out  C o  E (t ' )dt
t
0
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)
Step Input / Step Tracer
Experiment …
Cout  Co  E (t ' )dt '
t
0
Dividing by Co yields,
t
 Cout 

  0 E (t ' )dt '  F (t )
 Co  step
Differentiate this expression to obtain RTD function of E(t),
d  C (t ) 
E (t )  

dt  C o  step
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)
Step Input / Step Tracer
Experiment …
The +ve step is usually easier to carry out experimentally
than the pulse test, and it has the additional advantage that
the total amount of tracer in the feed over the period of the
test does not have to be known as it does in the pulse test.
Disadvantages for step input method;
a) Sometimes difficult to maintain a constant tracer conc in
the feed.
b) Differentiation of the data (lead to large errors)
c) Required large amount of tracer (expensive)
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)
Characteristics of the RTD…
E(t) sometimes is called as exit-age distribution function. It
characterizes the lengths of time various atoms spend at
reaction conditions.
RTD that commonly observed
RTD for Plug Flow Reactor
RTD for Near Perfectly Mixed CSTR
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)
Characteristics of the RTD…
RTD for Packed-Bed Reactor with Dead Zones & Channeling
Dead zones – serve to reduce the effective reactor volume, indicating
that the active reactor volume is smaller than expected.
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)
Characteristics of the RTD…
CSTR with dead zones
Tank reactor with shortcircuting flow (bypass)
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)
Characteristics of the RTD…
Integral Relationship
The fraction of the exit stream that has resided in the reactor for a
period of time shorter than a given value t is equal to the sum over all
times less than t of E(t)∆t, or expressed continuously,
 fraction _ of _ effluent

t
 which _ has _ been _ in _ reactor  F (t )
E
(
t
)
dt

0


 for _ less _ than _ time _ t

Analogously,
 fraction _ of _ effluent


 which _ has _ been _ in _ reactor  1  F (t )
E
(
t
)
dt

t


 for _ longer _ than _ time _ t

ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)
Characteristics of the RTD…
 fraction _ of _ effluent

t
 which _ has _ been _ in _ reactor  F (t )
E
(
t
)
dt

0


 for _ less _ than _ time _ t

Cumulative distribution function and called it F(t).
Can calculate F(t) at various time t from the area
under the curve of an E(t) vs t plot.
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)
Characteristics of the RTD…
The shape of the F(t) curve is shown for a tracer response to a step input
in figure below..
1.0
F(t)
0.8
80% [F(t)] of the molecules spend 40
min or less in the reactor and 20%
[1-F(t)] of the molecules spend longer
than 40min in the reactor
40
t
Cumulative distribution curve
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)
Mean Residence Time…
Ideal Reactor – parameter frequently used was SPACE TIME @
AVERAGE RESIDENCE TIME,  .
Ideal @ Non ideal Reactor – this nominal holding time, is equal to
mean residence time, t m
  tm
The mean value of variable is equal to the first moment of the RTD
function, E(t). Thus, the mean residence time is,

tm
tE(t )dt


  tE(t )dt
 E(t )dt

0

0
0
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)
Variance… 
2
Variance @ square of the standard deviation.
Is defined by,


2
   (t  t m ) E (t )dt
2
2
0
Is indication of spread of the distribution (greater value of variance, the
greater distribution’s spread)
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)
Skewness…s
Is defined by,
s 
3
1
 3/ 2
3


0
(t  t m ) 3 E (t )dt
Measures the extent that a distribution is skewed in one direction @
another in reference to the mean.
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)
Example: Mean Residence Time &
Variance Calculations
Calculate the mean residence time and the variance for the reactor
characterized in previous example by the RTD obtained from a pulse
input at 320K.
t(min)
0
1
2
3
4
5
6
7
8
9
10
12
14
C
(g/m3)
0
1
5
8
10
8
6
4
3
2.2
1.5
0.6
0
Solution;
The mean residence time, t m

  tE (t )dt
0
The area under the curve of plot of tE(t) as a function of t will yield tm.
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)
t(min)
0
1
2
3
4
5
6
7
8
9
10
12
14
C
0
1
5
8
10
8
6
4
3
2.2
1.5
0.6
0
E(t)
0
0.0
2
0.1
0.16
0.2
0.16
0.12
0.08
0.06
0.04
4
0.03
0.01
2
0
tE(t)
t-tm
(t-tm)2E(t)
To calculate tm we have to used integration formula in Appendix A.4 (text
book)using tE(t) data to get area under the curve of tE(t) VS t

10
14
0
0
10
t m   tE (t )dt   tE (t )dt   tE (t )dt
t m  5.15 min
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)
t(min)
0
1
2
3
4
5
6
7
8
9
10
12
14
C
0
1
5
8
10
8
6
4
3
2.2
1.5
0.6
0
E(t)
0
0.0
2
0.1
0.16
0.2
0.16
0.12
0.08
0.06
0.04
4
0.03
0.01
2
0
tE(t)
t-tm
(t-tm)2E(t)
To calculate variance, we use equation,

   (t  t m ) E (t )dt
2
2
0
Once u finished calculate this data for time 0 to 14min, we have to used
integration formula to get variance value.

   (t  t m ) E (t )dt   (t  t m ) E (t )dt   (t  t m ) 2 E (t )dt
2
2
0
  6.1min
  2.4 min
2
10
0
2
14
10
2
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)
Internal Ages Distribution, I(t)…
3
Is defined by,
1
1 t
I (t )  [1  F (t )]  1   E (t )dt 


  0
Is a fuction such that fraction of material inside the reactor.
It characterizes the time the material has been (and still is) in the reactor
at a particular time.
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)
RTD in Batch & Plug-Flow
Reactors
In Batch & PFR, all the atoms leaving such reactors have spent precisely
the same amount of time within the reactors.
The distribution function in such case is a spike of infinite height & zero
width, whose area is equal to 1.
The spike occurs at
t  V /  
or E (t )   (t   )

Mathematically, this spike is represented by the Dirac delta function:
E(t )   (t   )
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)
t

RTD in Batch & Plug-Flow
Reactors
The Dirac delta function has the following properties:
 (x) 
0
when x=0
∞
when x=0
Mean residence time is,

t m   tE (t )dt   t (t   )dt  
0
Variance is,

   (t   )  (t   )dt  0
2
2
0
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)
RTD in Batch & Plug-Flow
Reactors
The cumulative distribution function F(t) is,
F (t )    (t   )dt
t
0
in
out
1.0
E(t)
F(t)

t

PFR response to a pulse tracer input
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)
RTD in Single CSTR
In an ideal CSTR, the conc of any substances in the effluent stream is
identical to the conc throughout the reactor.
Use tracer balance to determine RTD for CSTR.
E(t) for CSTR,
Co e  t / 
e t / 
E (t )  
 

t / 

C
(
t
)
dt
C
e
dt

 o
C (t )
0
0
E ()  e

Where,

t

ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)
RTD for CSTR
The cumulative distribution function is,

F ( )   E ( )d  1  e 
0
1.0
E ( )
F ( )




CSTR response to a pulse tracer input
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)
RTD for CSTR
Mean residence time is,


0
0
t m   tE (t )dt  
Variance is,
 
2

0
t t / 
e dt  


(t   ) t / 
2
2 x
2
e dt    ( x  1) e dx  
0

2
 
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)
End For Part B
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)