ERT 208/4 REACTION ENGINEERING: Distribution of Residence Times for Reactors (PART B) By; Mrs Hafiza...
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Transcript of ERT 208/4 REACTION ENGINEERING: Distribution of Residence Times for Reactors (PART B) By; Mrs Hafiza...
ERT 208/4 REACTION
ENGINEERING: Distribution of
Residence Times for Reactors
(PART B) By; Mrs Hafiza Binti Shukor
ERT 208/4 REACTION ENGINEERINGSEM 2 (2009/2010)
ERT 208/4 REACTION ENGINEERINGSEM 2 (2009/2010)
C
C C
t t
Pulse injection
Step injection
Pulse response
Step response
MEASUREMENT OF MEASUREMENT OF RTD…RTD…2 most used methods of
injection :A) pulse inputB) step input
Step Input / Step Tracer Step Input / Step Tracer Experiment …Experiment …
ERT 208/4 REACTION ENGINEERINGSEM 2 (2009/2010)
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;
t
inout dttEttCtC0
)'()'()(
Step Input / Step Tracer Step Input / Step Tracer Experiment …Experiment …
ERT 208/4 REACTION ENGINEERINGSEM 2 (2009/2010)
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
)(tCo
0 t<0(Co) constant t>0
Step Input / Step Tracer Step Input / Step Tracer Experiment …Experiment …
ERT 208/4 REACTION ENGINEERINGSEM 2 (2009/2010)
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.
CinCout
t t
Step injection Step response
Because the inlet conc is a constant with time, Co we can take it outside the integral sign, that is
t
oout dttECC0
)'(
Step Input / Step Tracer Step Input / Step Tracer Experiment …Experiment …
ERT 208/4 REACTION ENGINEERINGSEM 2 (2009/2010)
Dividing by Co yields,
')'(0t
oout dttECC
)(')'(0
tFdttEC
C t
stepo
out
Differentiate this expression to obtain RTD function of E(t),
stepoC
tC
dt
dtE
)()(
Step Input / Step Tracer Step Input / Step Tracer Experiment …Experiment …
ERT 208/4 REACTION ENGINEERINGSEM 2 (2009/2010)
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)
Characteristics of the Characteristics of the RTD…RTD…
ERT 208/4 REACTION ENGINEERINGSEM 2 (2009/2010)
E(t) sometimes is called as exit-age distribution function. It characterizes the lengths of time various atoms spend at reaction conditions.
RTD for Plug Flow ReactorRTD for Near Perfectly Mixed CSTR
RTD that commonly observed
Characteristics of the Characteristics of the RTD…RTD…
ERT 208/4 REACTION ENGINEERINGSEM 2 (2009/2010)
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.
Characteristics of the Characteristics of the RTD…RTD…
ERT 208/4 REACTION ENGINEERINGSEM 2 (2009/2010)
CSTR with dead zones Tank reactor with short-circuting flow (bypass)
Characteristics of the Characteristics of the RTD…RTD…
ERT 208/4 REACTION ENGINEERINGSEM 2 (2009/2010)
Integral RelationshipThe 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,
)(
____
____
__
)(0
tF
ttimethanlessfor
reactorinbeenhaswhich
effluentoffraction
dttEt
)(1
____
____
__
)( tF
ttimethanlongerfor
reactorinbeenhaswhich
effluentoffraction
dttEt
Analogously,
Characteristics of the Characteristics of the RTD…RTD…
ERT 208/4 REACTION ENGINEERINGSEM 2 (2009/2010)
Cumulative distribution function and called it F(t).
)(
____
____
__
)(0
tF
ttimethanlessfor
reactorinbeenhaswhich
effluentoffraction
dttEt
Can calculate F(t) at various time t from the area under the curve of an E(t)
vs t plot.
Characteristics of the Characteristics of the RTD…RTD…
ERT 208/4 REACTION ENGINEERINGSEM 2 (2009/2010)
The shape of the F(t) curve is shown for a tracer response to a step input in figure below..
F(t)
t
1.0
0.8
40
Cumulative distribution curve
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
Mean Residence Mean Residence Time…Time…
ERT 208/4 REACTION ENGINEERINGSEM 2 (2009/2010)
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,
The mean value of variable is equal to the first moment of the RTD function, E(t). Thus, the mean residence time is,
mt
0
0
0 )()(
)(dtttE
dttE
dtttEtm
mt
Variance…Variance…
ERT 208/4 REACTION ENGINEERINGSEM 2 (2009/2010)
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)
2
0
22 )()( dttEtt m
2
Skewness…Skewness…
ERT 208/4 REACTION ENGINEERINGSEM 2 (2009/2010)
Is defined by,
Measures the extent that a distribution is skewed in one direction @ another in reference to the mean.
3s
0
3
2/3
3 )()(1
dttEtts m
Example: Mean Example: Mean Residence Time & Residence Time &
Variance Variance Calculations Calculations
ERT 208/4 REACTION ENGINEERINGSEM 2 (2009/2010)
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.
0
)( dtttEtm
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
ERT 208/4 REACTION ENGINEERINGSEM 2 (2009/2010)
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
min15.5mt
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 0.02
0.1 0.16 0.2 0.16 0.12 0.08 0.06 0.044
0.03 0.012
0
tE(t)
t-tm
(t-tm)2E(t)
10
0
14
100)()()( dtttEdtttEdtttEtm
ERT 208/4 REACTION ENGINEERINGSEM 2 (2009/2010)
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.
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 0.02
0.1 0.16 0.2 0.16 0.12 0.08 0.06 0.044
0.03 0.012
0
tE(t)
t-tm
(t-tm)2E(t)
0
22 )()( dttEtt m
14
10
210
0
2
0
22 )()()()()()( dttEttdttEttdttEtt mmm
min4.2
min1.6 22
Internal Age Internal Age Distribution, I(t)…Distribution, I(t)…
ERT 208/4 REACTION ENGINEERINGSEM 2 (2009/2010)
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.
3s
t
dttEtFtI0
)(11
)](1[1
)(
RTD in Batch & Plug-Flow RTD in Batch & Plug-Flow ReactorsReactors
ERT 208/4 REACTION ENGINEERINGSEM 2 (2009/2010)
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: )()( ttE
/Vt )()( ttE t
RTD in Batch & Plug-Flow RTD in Batch & Plug-Flow ReactorsReactors
ERT 208/4 REACTION ENGINEERINGSEM 2 (2009/2010)
The Dirac delta function has the following properties:
dtttdtttEtm )()(0
)(x0 when x=0
∞ when x=0
Mean residence time is,
Variance is,
0)()(0
22
dttt
RTD in Batch & Plug-Flow RTD in Batch & Plug-Flow ReactorsReactors
ERT 208/4 REACTION ENGINEERINGSEM 2 (2009/2010)
dtttFt
)()(0
The cumulative distribution function F(t) is,
E(t)
t
in
PFR response to a pulse tracer input
out
F(t)
1.0
RTD in Single CSTRRTD in Single CSTR
ERT 208/4 REACTION ENGINEERINGSEM 2 (2009/2010)
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,
eE )(
/
0
/
/
0)(
)()(
t
t
o
t
o e
dteC
eC
dttC
tCtE
t
Where,
RTD for CSTRRTD for CSTR
ERT 208/4 REACTION ENGINEERINGSEM 2 (2009/2010)
edEF 1)()(0
The cumulative distribution function is,
CSTR response to a pulse tracer input
1.0
)(F)(E
RTD for CSTRRTD for CSTR
ERT 208/4 REACTION ENGINEERINGSEM 2 (2009/2010)
dtet
dtttEt t
m 0
/
0)(
Mean residence time is,
Variance is,
0
222/
0
22 )1(
)( dxexdte
t xt
End For Part B
ERT 208/4 REACTION ENGINEERINGSEM 2 (2009/2010)