1. Distribution of Residence Times for Reactors

2. MEASUREMENT OF RTD…
2 methods of injection : A) pulse input – Already covered earlier B) step input
Pulse response
Pulse injection C
Step injection
Step response C C t t

3. Step Input :
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;

4. Step Input 0 t<0 (Co) constant t>0
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

5. Step Input
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

6. Step Input Dividing by Co yields,
Differentiate this expression to obtain RTD function of E(t),

7. Step Input
--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)

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

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.

10. Characteristics of the RTD…
CSTR with dead zones
Tank reactor with short-circuting flow (bypass)

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,

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.

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

14. Mean Residence Time…
Ideal Reactor – parameter frequently used was SPACE TIME as AVERAGE RESIDENCE TIME,
Ideal & 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,

15. Variance… Variance is square of the standard deviation. Is defined by,
Is indication of spread of the distribution (greater value of variance, the greater distribution’s spread)

16. Skewness… Is defined by,
Measures the extent that a distribution is skewed in one direction to another in reference to the mean.

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

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

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.

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

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:

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,

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

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,

25. RTD for CSTR The cumulative distribution function is,
1.0
CSTR response to a pulse tracer input

26. RTD for CSTR
Mean residence time is,
Variance is,