1. 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)

2. MEASUREMENT OF RTD… 2 most used methods of injection : A) pulse input
B) step input C
Pulse response
Pulse injection C
Step injection C
Step response t t
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)

3. 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;
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)

4. 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
t<0
(Co) constant t>0
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)

5. 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.
Cout
Cin
Step injection
Step response t t
Because the inlet conc is a constant with time, Co we can take it outside the integral sign, that is
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)

6. Step Input / Step Tracer Experiment …
Dividing by Co yields,
Differentiate this expression to obtain RTD function of E(t),
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)

7. 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;
Sometimes difficult to maintain a constant tracer conc in the feed.
Differentiation of the data (lead to large errors)
Required large amount of tracer (expensive)
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)

8. 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)

9. 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)

10. Characteristics of the RTD…
CSTR with dead zones
Tank reactor with short-circuting flow (bypass)
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)

11. 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,
Analogously,
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)

12. Characteristics of the RTD…
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)

13. 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)

14. Mean Residence Time…
Ideal Reactor – parameter frequently used was SPACE AVERAGE RESIDENCE TIME,
Non ideal Reactor – this nominal holding time, is equal to mean residence time,
The mean value of variable is equal to the first moment of the RTD function, E(t). Thus, the mean residence time is,
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)

15. Variance… Variance @ square of the standard deviation. Is defined by,
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)

16. Skewness… Is defined by,
Measures the extent that a distribution is skewed in one another in reference to the mean.
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)

17. 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.
Solution;
The mean residence time,
The area under the curve of plot of tE(t) as a function of t will yield tm.
t(min) 1 2 3 4 5 6 7 8 9 10 12 14
C (g/m3)
2.2
1.5
0.6
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)

18. t(min) 1 2 3 4 5 6 7 8 9 10 12 14 C
2.2
1.5
0.6
E(t)
0.02
0.1
0.16
0.2
0.12
0.08
0.06
0.044
0.03
0.012
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
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)

19. To calculate variance, we use equation,
t(min) 1 2 3 4 5 6 7 8 9 10 12 14 C
2.2
1.5
0.6
E(t)
0.02
0.1
0.16
0.2
0.12
0.08
0.06
0.044
0.03
0.012
tE(t)
t-tm
(t-tm)2E(t)
To calculate variance, we use equation,
Once u finished calculate this data for time 0 to 14min, we have to used integration formula to get variance value.
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)

20. Internal Age Distribution, I(t)…
Is defined by,
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)

21. 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 or
Mathematically, this spike is represented by the Dirac delta function:
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)

22. RTD in Batch & Plug-Flow Reactors
The Dirac delta function has the following properties:
when x=0
∞ when x=0
Mean residence time is,
Variance is,
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)

23. RTD in Batch & Plug-Flow Reactors
The cumulative distribution function F(t) is, 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)

24. 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,
Where,
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)

25. RTD for CSTR The cumulative distribution function is,
1.0
CSTR response to a pulse tracer input
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)

26. RTD for CSTR Mean residence time is, Variance is,
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)

27. End For Part B
ERT 208/4 REACTION ENGINEERING
SEM 2 (2009/2010)