5 Reactor Vessels - MIT OpenCourseWare Reactor Vessels Motivation Fully-mixed reactors ......

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5 Reactor Vessels Motivation Fully-mixed reactors Single & multiple tanks with pulse, step and continuous inputs Dispersed flow reactors with pulse and continuous inputs Examples

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

Wastewater Treatment Plants

Secondary settling

Primary settling

Waste stabilization

Constructed Wetlands

Cell 6

Cell 7

Cell 5

Cell 12

Potash Evaporation Ponds

Southern shoreline Dead Sea

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)

Benthic Flux Chambers

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?

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 tank

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)