5 Reactor Vessels - MIT OpenCourseWare Reactor Vessels Motivation Fully-mixed reactors ......
Transcript of 5 Reactor Vessels - MIT OpenCourseWare Reactor Vessels Motivation Fully-mixed reactors ......
5 Reactor Vessels
MotivationFully-mixed reactorsSingle & multiple tanks with pulse, step and continuous inputsDispersed flow reactors with pulse and continuous inputsExamples
Introductionm&
Qin Qout
A tank, reservoir, pond or reactor with controlled in/out flow
If Qin = Qout = Q then tres = τ = t* = V/Q
Used interchangeablyGenerally more interest in what comes out than what’s inside reactor
Applications
Natural ponds and reservoirsEngineered systems (settling basins, constructed wetlands, combustion facilities, chemical reactors, thermo-cyclers…)Laboratory set-ups: simple configurations with known mixing so you can predict/control the fate processes
Classification
Flow: continuous flow or batchSpatial structure: well-mixed or 1-, 2-, 3-DLoading: continuous, intermittent (step) or instantaneous (pulse)Single reactor or reactors-in-series
Initially look at single continuously stirred tank reactor (CSTR)
Well-mixed tank, arbitrary input
c
cinincQm =&
t t
)()( tQtQdtdV
outin −= Conservation of volume
VrtQtctQtcdtcVd
ioutinin +−= )()()()()(Conservation of mass
Well-mixed tank
c
cinincQm =&
kct
ctcdtdc in −
−=
*)(
ttt−τ
If Qin = Qout = const, V = const; t* = V/Q; and ri = -kc
[ ]∫ −+−−+− +=*/
0
)(*/)()*/( *)/()()(tt
tkttin
kttto tdecectc ττ ττ
Convolution integral (exponential filter that credits inputs w/ decreasing weight going backwards in time)
IC
Roadmap to solutions (3x2)inincQm =& inincQm =& inincQm =&
Inputs
t t t
Pulse Step Continuous
Reactors
Single Multiple
Pulse input to single tankinincQm =&
Pulse
May have practical significance—e.g. instantaneous spillMore commonly used as a diagnostic; Produces stronger gradients than other types of inputs
Pulse input, single tank, cont’dinincQm =&
)*/()( kttto eV
Mtc +−=Mo
co = Mo/V (not initial conc.)Pulse
)*/()( kttt
o
ec
tc +−=
Know t* and co;
Plot c/co vs t/t*
Determine kt* = > k
WE 5-2
Pulse input, single tank, cont’dinin cQm =&
Pulse
Special cases:General case:
No flow; batch reactor
kt
o
ec
tc −=)(
)*/()( kttt
o
ec
tc +−=No loss; just flushing
VtQtt
o
eectc /*/)( −− ==
In practice never exactly well-mixed
Data versus theory tells how close to well mixedWays to mix
Directed discharge and intakeBafflesStirrers
May need to avoid over-mixing (benthic flux chambers)
White, (1974) Figure by MIT OCW.
0
400
7 March8 March 1973
Perfectly mixed system
Cou
nt R
ate
with
Flo
atin
g D
etec
tor
in
Out
let B
ay (c
ount
s/s)
800
1200 4
Time from Injection of Tracer (h)
8 12 16 20 24 28
16 20 4 8 12 16 20Mid Night
Time of Day (hrs)
Slight diversion: Residence time properties of reactor vessels
Not necessarily well-mixed
But known residence time
c/co
t/t*
1
1
Co = Mo/V t* = V/Q cV/Mo=coutV/Mo
tQ/V
inincQm =&
Mo
t
Residence time distribution
Unit impulse response
Recall from Chapter 4m& m* f* = -dm*/dt = coutQ
0 t 0 t 0 t
11
Rate of injection Mass remaining in system Mass leaving rate
**
1/*
tcc
VQ
ccQcf
VVMc
QVt
o
out
o
outout
oo
===
==
=
1**
1**1**
1*
1*
0
00
00
=⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛⇒
==
=⎟⎠⎞
⎜⎝⎛⇒=
∫
∫∫
∫∫
∞
∞∞
∞∞
ttd
tt
cc
tdtft
orttdtf
ttd
cc
dtf
o
out
o
outOthMoment
1st Moment
RTD, cont’d0th and 1st moments of normalized distribution (cout/co vs t/t*) are both unity
For well mixed tank, k=0 (CSTR)
1*)/(*)/(
1*)/(
*/
*/
*/
=
=
==
∫
∫∞
−
∞−
−
o
tt
o
tt
tt
oo
out
ttdtte
ttde
ecc
cc
c/co
1
1 t/t*Area under curve and center of mass are both one
Pulse input to tanks-in-seriesinincQm =&
)*/(1
*)!1(kttt
nn
o
ent
tnn
cc +−
−
⎥⎦⎤
⎢⎣⎡
−=
iiii kc
tcc
dtdc
−−
= −
*1
t
Co = Mo/nV; t* = V/Q = res. time of each tank; total res time = nt*
Mo
V
2.0
1.0
0.0 .5 1.0 1.5 2.0
Dimensionless Time, t/nt*
Fully-Mixed Tank
n = 1
2
5
10
15
20
n = 40Plug Flow
Reactor
.k = 0a = 0
∞n = ∞
C
C0
Figure by MIT OCW.
Pulse input to tanks-in-series
Area under curve and center of mass are both one
1**
*)/(
0
=⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛∫
∞
nttd
ntt
cnttc
o
1*
*)/(
0
=⎟⎠⎞
⎜⎝⎛∫
∞
nttd
cnttc
o
Limits of plug flow and fully well-mixed
2.0
1.0
0.0 .5 1.0 1.5 2.0
Dimensionless Time, t/nt*
Fully-Mixed Tank
n = 1
2
5
10
15
20
n = 40Plug Flow
Reactor
.k = 0a = 0
∞n = ∞
C
C0
Figure by MIT OCW.
Advantages of Plug Flow?
e-kt (this function multiplies k=0 sol’n)1
t/nt*10.5 1.5
Everything “cooks” the same time. The mean residence time of a water parcel is always nt* (by definition) but under plug flow all parcels reside for nt*
This advantage obviously applies for continuous and step injections as well
Step input, single and multipleinin cQm =&
tStep
[ ])*/()*/( 1*1
)( ktttinkttto e
ktc
ectc +−+− −+
+=
⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−
++++−=−
−
)!1()(...
2)(11
)*(1)( 12
nttte
tctc n
tn
in
κκκκ
κ
*/1 tk +=κ
Multiple reactors, k = 0
0.1
0.2
0.2 0.4 0.6 0.8
0.3
0.4
0.5
0.6
0.7
0.8
0.9
00
1
1 1.2 1.4 1.6 1.8 2
Normalized time, t/ntstar
Nor
mal
ized
con
cent
ratio
n, c
/cin
Response of n tanks to step input, n = 1, 2, 4, 8, and 16
n = 116
Figure by MIT OCW.
WE 5-3 Thermal storage tankDay
T2
Night
Collector
DemandT1
t = start of the “day”) (when tank has cold water at T = T1 and you want to keep T = T1⎥
⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−
++++−=−− −
−
)!1()(...
2)(11
)*(1)( 12
12
1
nttte
tTTTtT n
tn
κκκκ
κ
Percent of theoretical cooling (or heating) potential, a measure of hydraulic efficiency of storage system
dtT
tTTnt
tRt
∫ ∆−
=0
0
2 )(*%100)(
Thermal storage tank, cont’d
100
50
00 1.0
m = ∞ (plug - flow)
m = l (well - mixed)
m = 5
Non - dimensional Time t / (V/Q)
4 Horiz. Barriers4 Vert. BarriersNo Barriers
% o
f tot
al th
eore
tical
coo
ling
pote
ntia
l rec
over
ed
Figure by MIT OCW. Geyer and Golay (1983),Adams (1986)
Continuous input single & multiple
inin cQm =&
From step input solutions for large t
Continuous
*11ktc
c
in +=
nin
out
ktcc
*)1(1
+=
Continuous input single & multiple
nin
out
ktcc
*)1(1
+= (knt* = 1; constant total volume)
n kt* cout/cin
1 1 0.5
2 0.5 0.44
5 0.2 0.40
10 0.1 0.386
100 0.01 0.37
infinity 0 e-1
Dispersed flow reactor
W
x L
Engineer delayed response (--> plug flow) by making the reactor long & narrow. Density stratification should also be minimal (see Sect 5.4.1)
kcxcAE
xAxcu
tc
L −⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
=∂∂
+∂∂ 1
(Small) entrance and exit sections may be treated differently
Pulse input
Qin = Qout = const; A = const; k = 0 and pulse input at x =0
∑∞
=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
⎟⎠⎞
⎜⎝⎛⎟
⎟⎠
⎞⎜⎜⎝
⎛+
−
⎟⎟⎠
⎞⎜⎜⎝
⎛++
⎟⎠⎞
⎜⎝⎛ +
=1
22
22 *
42
exp
4
cossin22
i
i
i
iiii
o
L
tt
Pe
PePe
PePe
Pe
cc
µ
µ
µµµµ
⎟⎟⎠
⎞⎜⎜⎝
⎛−= −
i
ii
PeP µµ
µ4
cot 1co = Mo/V; Pe = UL/EL
Dispersed flow, pulse input
A lot like tanks-in-seriesGreater n or Pe(lower EL) => plug flowPe ~ 2n-1 ( WE 5-5)Elongation sometimes done with baffles
2.0
1.0
00 .5 1.0 1.5 2.0
Fully-Mixed Tank
2
∞ p = ∞
C0
CL
Plug Flow Reactor
4610
15
25
35
2.8
t/t*
P = 0
Figure by MIT OCW.
Dispersed flow vs tanks-in-series
2.0
1.0
00 .5 1.0 1.5 2.0
Fully-Mixed Tank
2
∞ p = ∞
C0
CL
Plug Flow Reactor
4610
15
25
35
2.8
t/t*
P = 0
2.0
1.0
0.0 .5 1.0 1.5 2.0
Dimensionless Time, t/nt*
Fully-Mixed Tank
n = 1
2
5
10
15
20
n = 40Plug Flow
Reactor
.k = 0a = 0
∞n = ∞
C
C0
Figure by MIT OCW. Figure by MIT OCW.
Dispersed flow reactor
Area under curve and center of mass again are both one
1*
*)/(
0
=⎟⎠⎞
⎜⎝⎛∫
∞
nttd
cnttc
o
1**
*)/(
0
=⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛∫
∞
nttd
ntt
cnttc
o
Limits of plug flow and fully well-mixed
2.0
1.0
00 .5 1.0 1.5 2.0
Fully-Mixed Tank
2
∞ p = ∞
C0
CL
Plug Flow Reactor
46
10
15
25
35
2.8
t/t*
P = 0
Figure by MIT OCW.
(Idealized) effect of central baffle
2
1BHE
QAEB
AHBQ
EULPe
LLL
=== If EL = const, B decreases by 2x, so Pe increases by 4x; real increase could be even more
22
2
*
22
~~
~01.0
BLH
UBULH
EULPe
HUB
HuBUE
L
L
=
≅L increases by 2x and B decreases by 2x, so Pe increases by 8x.
Can you place the UIR (A-D) with the schematic cooling pond (I-IV)?
Cerco, 1977
2
1
00 0.5 1 tQ
V
1
00 0.5 1 tQ
V
1
00 0.5 1 tQ
V
2
3
1
00 0.5 1 tQ
V
0.5
0.5
CVMo
CVMo
CVMo
CVMo
I
II
III
IV
D
C
B
A
Figure by MIT OCW.
WE5-5 Dispersed flow vstanks-in-series
River: L = 10 km; B = 20 m; H = 1 m; Q = 10m3/s; S = 10-4
What are equivalent values of Pe and n?
smgHSu /032.010110 4* =⋅⋅=≅ −
u = Q/BH =0.5 m/s
*22 /01.0 HuBUEL = sm /30)032.01/(205.001.0 222 ≅⋅⋅⋅≅
16730/105.0/ 4 =⋅== LEULPe
.832/ =≅ PnA finite difference model using upwind differencing would want to use a spatial grid size of order 104/83 or 120 m
Dispersed flow, continuous input
W
kcdx
cdEdxdcu L −= 2
2
+− == ⎟⎠⎞
⎜⎝⎛ −= 00 xLxin dx
dcAEQcQc
GE assuming steady state
at x = 0 (Type III)
+− === LxLx QcQc at x = L (Type II)
Boundary conditions
x L
Dispersed flow, continuous inputSolution (0 < x < L)
( ) ( ) ( )⎭⎬⎫
⎩⎨⎧
⎥⎦⎤
⎢⎣⎡ −−−⎥⎦
⎤⎢⎣⎡ −+⎟
⎠⎞
⎜⎝⎛= 1
2exp11
2exp)1(
2exp)( ξααξααξ PePePeg
cxc
oin
Lx /=ξ
( ) ( ) ⎟⎠⎞
⎜⎝⎛−−−⎟
⎠⎞
⎜⎝⎛+
=
2exp1
2exp1
222 PePe
go αααα
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛+=
PeukLa 141
Solution (0 < x < L)
Plug flow: greater loss at small x lower concentration at large x
Note:
c(x=0) < cin
At outlet
( )2/exp2 Peagcc
oin
L =
Solution (x=L)
For Pe --> 0 (well mixed)
*11ktc
c
in
L
+=
For Pe --> oo (plug flow)
*/ ktukL
in
L eecc −− ==
WE 5-6 Continuous solar disinfection
SODIS: simple household treatment technology for destroying pathogens using UV and temperaturePioneered by EAWAG, SANDEC and others; studied by former MEng students in NepalContinuous (or semi-continuous) operation more convenient than discrete bottles
Dispersed flow reactor response to step input (eq 2.104), k = 0
Cout/Cin vs t/t* for various values of Pe (2, 4, 8, 16, 32, 64, 128) compared with measurements
(connected dots)
Pe= 2
Pe = 128
Xanat’s SC-SODIS
Infer 16 < Pe < 128
Flores (2003)
Continuous SODIS, cont’dMeasurements show 99% of pathogens killed for t* = 2 days (cout/cin = 0.01). Use Eq. 5.40 to estimate first order removal rate (k) for plausible values of Pe & compare with plug flow reactor
0.01
Example
Pe = oo, cL/cin = cout/cin = 0.01
kt* = 4.6, k = 2.3 d-1
Continuous SODIS, cont’d
Pe k (d-1) Cout/cin
(Plug Flow)Cout (Dispersed)Cout (Plug flow)
16 2.92 0.0029 3.45
32 2.62 0.0053 1.89
64 2.47 0.0072 1.39
128 2.38 0.0086 1.16
oo 2.30 0.0100 1.00
Selective withdrawal of water from density stratified tank or reservoir
Withdrawal of lower layer water tends to “suck down” upper layer. Bernoulli’s equation used to compute max flow, draw down
Skimmer walls
21
3
32
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛′= rc hgq
Maximum flow per unit width
( )2/3
3 32⎟⎠⎞
⎜⎝⎛=
′=
r
rhg
qF
For larger Fr, partially effective
W Intake Channel
Plan View
Skimmer Wall
Skimmer Wall
Design Condition For Full Effectiveness
Partially Effective Condition
Critical Section
Bq
q1
q2
Mixing H
ρ
ρ
ρ + ∆ρ
ρ + ∆ρ
hc1
hc2
hr
hr
Figure by MIT OCW.
Partially-effective skimmer wall
0.6
0.5
0.4
0.3
0.2
0.1
0.00.0 0.4 0.8 1.2 1.6 2.0
λ
λ =
q 1/q
H/hr = 0.0
H/hr = 1.1
3.02.5
2.0
1.8
1.6
1.4
1.2
λ = 0.99 λmax
λmax = 1-H/hr
1
λ = 1− Frc/Fr
Frc = (2/3)3/2
Fr = q (g' hr3)-1/2
H/hrDataHorleman andElder (1965)Delft Laboratory(1973)State ElectricityCommission ofVictoria (1973-74)
2.6 to 7.4
3.0 to 6.0
1.091.651.7
1.71.751.65
1.61.2
Figure by MIT OCW.Jirka (1979)