Dispersed flow reactor
response to spike input
Pe = ∞
c
c0
t/tR
Fraction remaining
Dispersed-flow reactor performance for k = 0.5/day
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Pe = 0 (FMT)
Pe = 1
Pe = 2
Pe = 10
Pe = ∞ (PFR)
0
2
4
6
Residence time (days)
8
10
Dispersed-flow reactor performance for k =
0.5/day
Fraction remaining
1.000
0.100
Pe = 0 (FMT)
Pe = 1
Pe = 2
Pe = 10
Pe = ∞ (PFR)
0.010
0.001
0
2
4
6
Residence time (days)
8
10
Dimensionless concentration, c/c0
3
n=∞
40
2.5
2
20
1.5
10
5
1
2
0.5
n=1
0
0
0.5
1
Dimensionless time, t/tR
1.5
2
Tanks-in-series compared to dispersed flow reactor
Dispersed flow
Tanks-in-series
Tanks-in-series with exchange flow
c
c0
t/tR
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
Q
V
Q
Measure dye
concentration at
outlet
Inject slug of dye
at inlet at t=0
RTD( t) =
C( t)
(for steady flow)
∞
∫ C(t)dt
0
Note units - RTD is in inverse time
∞
by definition:
∫ RTD(t)dt = 1
(i.e., total probability = 1)
0
∞
tD = ∫ t RTD( t )dt = first moment of RTD = tracer detention time
0
Plug flow
RTD = δ(t-tR)
tR•RTD
1
CSTR
RTD = 1/tR exp(-t/tR)
0.38
0.14
tR
2tR
t
RTD math:
Dirac delta function (or unit impulse function)
15
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( t )dt = 1
−∞
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
∞
∞
0
0
∫ RTD(t )dt = ∫ δ(t − t
R
)dt = 1 zeroth moment
Note lower limit is 0 and not -∞ since you can’t have negative
residence time
(i.e., fluid leaving before it entered)
∞
∞
0
0
tD = ∫ t RTD( t )dt = ∫ t δ( t − tR )dt = tR
first moment (mean) =
tracer detention time
CFSTR
RTD(t) = exp(-t/tR) / tR units of t-1
∞
∞
∞
⎡ exp( − t / t R ) ⎤
⎡
exp( − t / t R )
exp(0) ⎤
RTD
(
t
)
dt
dt
t
t
0
=
=
−
=
−
−
R
R
⎢
⎥
⎢
⎥ =1
∫0
∫0 t R
t
t
R
R
⎣
⎦0
⎣
⎦
∞
⎤
exp( −t / tR )
1 ⎡ exp( − t / tR )
(− t / tR − 1)⎥
tD = ∫ t RTD( t )dt = ∫ t
dt = ⎢
2
tR
tR ⎣ 1 tR
0
0
⎦0
= tR [0 − exp(0)(− 0 − 1)] = tR
∞
∞
Note: from CRC Tables:
∫
xe ax dx =
e ax
( ax − 1 )
a2
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
16
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)
t2
t1
t3
t4
Exit concentration:
Ce = C0 exp(-kt4)
Consider a different parcel, taking a longer route:
Exit concentration Ce=C0 exp(-kt6) where t6 > t4
t2
t1
t3
t4
t2
t3
t4
t5
t6
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:
∞
∫
C e = RTD( t ) C 0 exp( −kt )dt
0
At t1 Ce = C0 exp(-kt1)
RTD
At t2 Ce = C0 exp(-kt2)
t1
t
t2
17
Residence Time Distribution for Real Systems
QR
CI
At inlet
Recirculation
QR
Ce
At outlet
Real circulation has:
Short circuiting
Dead zones (exclusion zones)
RTD from tracer study ≠ plug flow or stirred tank reactor
RTD
tD
t
tR
Detention time, TD
∞
tD =
∫ tRTD(t )dt
0
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
∞
σ = ∫ ( t − t D ) 2 RTD( t ) dt
2
0
⎛σ
As a dimensionless number, d = ⎜⎜
⎝ tD
⎞
⎟⎟
⎠
2
RTD
σ
t
As σ → 0, no mixing, plug flow
As σ → ∞, complete mixing, CFSTR
18
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!
19
Lecture 4.doc
Figure by MIT OCW.
Adapted from: Camp, T. R. "Sedimentation and the design of settling tanks."
Transactions ASCE 111 (1946): 895-936.