ocw.mit.edu · pdf filetransactions asce 111 (1946): 895-936. created date: 1/31/2005 1:29:56...
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Pe = ∞
0cc
t/tR
Dispersed flow reactorresponse to spike input
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Dispersed-flow reactor performance for k = 0.5/day
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Frac
tion
rem
aini
ng
Pe = 0 (FMT) Pe = 1 Pe = 2 Pe = 10 Pe = ∞ (PFR)
0 2 4 6 8 10
Residence time (days)
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Dispersed-flow reactor performance for k = 0.5/day
0.001
0.010
0.100
1.000
0 2 4 6 8 Residence time (days)
Frac
tion
rem
aini
ng
Pe = 0 (FMT) Pe = 1 Pe = 2 Pe = 10 Pe = ∞ (PFR)
10
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0
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5 2Dimensionless time, t/tR
Dim
ensi
onle
ss c
once
ntra
tion,
c/c
0
n=1
2
5
10
20
40n=∞
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Tanks-in-series compared to dispersed flow reactor
Dispersed flow Tanks-in-series
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Tanks-in-series with exchange flow
t/tR
0cc
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15
Residence Time Distributions
We have seen two extreme ideals: Plug Flow – fluid particles pass through and leave reactor in same
sequence in which they enter Stirred Tank Reactor – fluid particles that enter the reactor are
instantaneously mixed throughout the reactor
Residence time distribution - RTD(t) – represents the time different fractions of fluid actually spend in the reactor, i.e. the probability density function for residence time
∫∞
=
0
dt)t(C
)t(C)t(RTD (for steady flow)
Note units - RTD is in inverse time
by definition: ∫∞
=0
1dt)t(RTD (i.e., total probability = 1)
∫∞
=0
D dt)t(RTDtt = first moment of RTD = tracer detention time
RTD math: Dirac delta function (or unit impulse function)
V Q Q
Measure dye concentration at outlet
Inject slug of dye at inlet at t=0
tR•RTD
tR t
CSTR RTD = 1/tR exp(-t/tR)
Plug flow RTD = δ(t-tR)
1
0.14
0.38
2tR
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16
Represents a unit mass concentrated into infinitely small space resulting in an infinitely large concentration
δ(t) = ∞ at t = 0, 0 at t ≠ 0
∫∞
∞−
=δ 1dt)t(
Can think of Dirac delta function as extreme form of Gaussian M0δ(t-τ) is spike of mass M0 at time τ
Plug Flow RTD(t) = δ(t-tR) with implied units of t-1
∫ ∫∞ ∞
=−δ=0 0
R 1dt)tt(dt)t(RTD zeroth moment
Note lower limit is 0 and not -∞ since you can’t have negative residence time (i.e., fluid leaving before it entered)
∫ ∫∞ ∞
=−δ==0
R0
RD tdt)tt(tdt)t(RTDtt first moment (mean) =
tracer detention time CFSTR
RTD(t) = exp(-t/tR) / tR units of t-1
1t
)0exp(0tt
)t/texp(tdtt
)t/texp(dt)t(RTDR
R0R
RR
0 R
R
0
=⎥⎦
⎤⎢⎣
⎡−−=⎥
⎦
⎤⎢⎣
⎡ −−=
−=
∞∞∞
∫∫
( )
( )[ ] RR
0
R2R
R
R0 R
R
0D
t10)0exp(0t
1t/tt1
)t/texp(t1dt
t)t/texp(tdt)t(RTDtt
=−−−=
⎥⎦
⎤⎢⎣
⎡−−
−=
−==
∞∞∞
∫∫
Note: from CRC Tables: ∫ −= )ax(aedxxe
axax 1
2
Control Volume Models and Time Scales for Natural Systems
What are actual systems like? Plug flow or Stirred reactor It depends upon the time scales:
Mixing time for plug flow reactor is infinite: it never mixes Mixing time for stirred reactor is zero: it mixes instantaneously
When are these assumptions realistic? We need to estimate the time of the real system to mix - tMIX
compared to time to react If tMIX << tR → stirred reactor If tMIX >> tR → plug flow reactor
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17
Residence Time and Reactions
RTD provides a means to estimate pollutant removal Consider a 1st-order reaction: C(t) = C0 exp(-kt) This reaction applies to any water mass entering and exiting the system –
view from Lagrangian perspective (i.e., following the parcel of water)
Exit concentration: Ce = C0 exp(-kt4)
Consider a different parcel, taking a longer route: Exit concentration Ce=C0 exp(-kt6) where t6 > t4
If a plug flow model applies, the exit concentration is simple: all parcels exit at exactly TR
In a natural system, it is not perfect plug flow, therefore look at RTD RTD gives the probability that the fluid parcel requires a given amount of
time to pass system On average:
dt)ktexp(C)t(RTDC0
0e −= ∫∞
At t1 Ce = C0 exp(-kt1) At t2 Ce = C0 exp(-kt2)
t1
t2
t3t4
RTD
t t1 t2
t1
t2t3
t4t2t3 t4
t5t6
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18
Residence Time Distribution for Real Systems
Real circulation has: Short circuiting Dead zones (exclusion zones)
RTD from tracer study ≠ plug flow or stirred tank reactor
Detention time, TD
tD = ∫∞
0
dt)t(tRTD
Note distinction with hydraulic residence time, tR = V/Q tD = tR if and only if there are no exclusion zones
Variance of RTD is a measure of mixing
∫∞
−=σ0
2D
2 dt)t(RTD)tt(
As a dimensionless number, 2
Dtd ⎟⎟
⎠
⎞⎜⎜⎝
⎛ σ=
As σ → 0, no mixing, plug flow As σ → ∞, complete mixing, CFSTR
QR CI At inlet
QR Ce At outlet
Recirculation
RTD
ttD tR
RTD
t σ
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19
Residence Time Distribution for Real Systems
Review some concepts: Two models for mixing
Plug flow Stirred reactor
Time scales: tR = V/Q mean hydraulic residence time (nominal residence time) tREACTION =1/k (or for 95% complete reaction or removal 3/k) tADV = L/u
Limitations of tR in describing residence times of true systems because of dead zones, recirculation, short circuiting
Consider alteration of the real system: Add berms to control circulation!
Lecture 4.doc
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Figure by MIT OCW.
Adapted from: Camp, T. R. "Sedimentation and the design of settling tanks."Transactions ASCE 111 (1946): 895-936.