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Chapter 5 Chapter 5 Bistability and oscil Bistability and oscil lations lations in flow reactors in flow reactors Closed system: batch From non-equilibrium to equilibriu Closed system: batch From non-equilibrium to equilibriu m Open system: to maintain non-equilibrium state reacta Open system: to maintain non-equilibrium state reacta nts flow in and products flow out nts flow in and products flow out If the reactor is well stirred, we call it CSTR(continuou If the reactor is well stirred, we call it CSTR(continuou s-flow stirred tank reactor) s-flow stirred tank reactor) Molecules can spend different time Molecules can spend different time at the CSTR because of flowing. at the CSTR because of flowing. The average time spent in reactor The average time spent in reactor is called the mean residence time is called the mean residence time T T res res = volume/ flow rate. = volume/ flow rate. The system can display steady state, bistable state, oscil The system can display steady state, bistable state, oscil lations, chaos in a CSTR. lations, chaos in a CSTR. http://www.grc.org/programs.aspx?year http://www.grc.org/programs.aspx?year =2008&program=oscillat =2008&program=oscillat reacta nts stirre r

Chapter 5 Bistability and oscillations in flow reactors

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reactants. stirrer. Chapter 5 Bistability and oscillations in flow reactors. Closed system: batch From non-equilibrium to equilibrium Open system: to maintain non-equilibrium state reactants flow in and products flow out - PowerPoint PPT Presentation

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Page 1: Chapter 5        Bistability and oscillations              in flow reactors

Chapter 5 Chapter 5 Bistability and oscillations Bistability and oscillations in flow reactors in flow reactors

Closed system: batch From non-equilibrium to equilibriumClosed system: batch From non-equilibrium to equilibriumOpen system: to maintain non-equilibrium state reactants flow in and prOpen system: to maintain non-equilibrium state reactants flow in and pr

oducts flow outoducts flow outIf the reactor is well stirred, we call it CSTR(continuous-flow stirred tank reaIf the reactor is well stirred, we call it CSTR(continuous-flow stirred tank rea

ctor)ctor) Molecules can spend different time Molecules can spend different time at the CSTR because of flowing. at the CSTR because of flowing. The average time spent in reactor The average time spent in reactor is called the mean residence time is called the mean residence time TTresres= volume/ flow rate.= volume/ flow rate.The system can display steady state, bistable state, oscillations, chaos in a The system can display steady state, bistable state, oscillations, chaos in a

CSTR.CSTR.

http://www.grc.org/programs.aspx?year=2008&http://www.grc.org/programs.aspx?year=2008&program=oscillatprogram=oscillat

reactantsstirrer

Page 2: Chapter 5        Bistability and oscillations              in flow reactors

5.1 Steady state and bistability5.1 Steady state and bistability

Example: Iodate - arsenite system Example: Iodate - arsenite system IOIO33

--+5I+5I--+3H3AsO+3H3AsO33→6 I→6 I--+3 H3AsO+3 H3AsO44

At high flow rate, the concentrations of reactants in a CSTR have noAt high flow rate, the concentrations of reactants in a CSTR have not bigger different with the inflow concentrations, this branch is kt bigger different with the inflow concentrations, this branch is known as nown as flow branchflow branch(( 流动分枝 流动分枝 ).).

At low flow rate, the system in a CSTR approach the thermodynamiAt low flow rate, the system in a CSTR approach the thermodynamic equilibrium , the curve is called c equilibrium , the curve is called thermodynamic branch(thermodynamic branch( 热力热力学分枝学分枝 ))

As the flowrate is decreased, the system changes from flow branch tAs the flowrate is decreased, the system changes from flow branch to thermodynamic brancho thermodynamic branch

Attention: when flowrate change, the system evolve to new state fAttention: when flowrate change, the system evolve to new state for short(low flowrate) or long (high flow rate)time.or short(low flowrate) or long (high flow rate)time.

reactants stirrer

Page 3: Chapter 5        Bistability and oscillations              in flow reactors

Situations:Situations:1. When changing flowrate up or down, the plot of steady concentra1. When changing flowrate up or down, the plot of steady concentra

tions have only one curve. But during the some region of flowrattions have only one curve. But during the some region of flowrate, there is a quick change of concentration from one branch to ane, there is a quick change of concentration from one branch to another branch. other branch. flow diagramflow diagram

Page 4: Chapter 5        Bistability and oscillations              in flow reactors

2. Thermodynamic branch and flow branch overlap as the direction 2. Thermodynamic branch and flow branch overlap as the direction of changing flowrate is opposite. This is phenomena is named bisof changing flowrate is opposite. This is phenomena is named bistabilty. Within the region of bistability, the actual state selected dtabilty. Within the region of bistability, the actual state selected depends on not only the parameter but also the operational historepends on not only the parameter but also the operational history.y.

When the flowrate is up or down, the state jumps or fall off in delaWhen the flowrate is up or down, the state jumps or fall off in delay. This phenomena is called hysteresis y. This phenomena is called hysteresis

This abrupt changes in compositions is discontinuous responded tThis abrupt changes in compositions is discontinuous responded to the continuous changes in the operating conditions. This is bifuo the continuous changes in the operating conditions. This is bifurcation.rcation.

Page 5: Chapter 5        Bistability and oscillations              in flow reactors

5.2 Dynamic equations for flow reactors5.2 Dynamic equations for flow reactors

The changes of concentrations in flow reactor result froThe changes of concentrations in flow reactor result from the net flow and reactionm the net flow and reaction

V dA/dt=q(AV dA/dt=q(A00-A)+VR-A)+VRdA/dt=k0(A0-A)+R kdA/dt=k0(A0-A)+R k00=q/V time=q/V time-1-1

For iodate-arsenite system For iodate-arsenite system

IO3-+5I-+6H+=6I-+3H2OIO3-+5I-+6H+=6I-+3H2O

R=(KR=(Ka1a1+K+Ka2a2[I-])[I-][ IO[I-])[I-][ IO33--][H+]][H+]22 Mixed autocalysis Mixed autocalysis

d[IOd[IO33-]/dt=k0([[IO-]/dt=k0([[IO33-]-]00-[IO-[IO33-])- (Ka1+Ka2[I-])[I-][ IO3-][H+]-])- (Ka1+Ka2[I-])[I-][ IO3-][H+]22

Page 6: Chapter 5        Bistability and oscillations              in flow reactors

5.3 Steady state solutions: flow diagrams5.3 Steady state solutions: flow diagramsConsidering cubic autocatalysis only in iodate-arsenite systemConsidering cubic autocatalysis only in iodate-arsenite system

d[IOd[IO33--]/dt=k]/dt=k00([[IO([[IO33-]-]00-[IO-[IO33-])- (Ka1+Ka2[I-])[I-][ IO3-][H+]-])- (Ka1+Ka2[I-])[I-][ IO3-][H+]2 2

Ka1=0Ka1=0 kkcc=ka2[H=ka2[H++]2]2

d[IOd[IO33-]/dt=k0([[IO-]/dt=k0([[IO33--]]00-[IO-[IO33

--])- ])- kkcc [I [I--]]22[ IO3-][ IO3-] conservation of element iodineconservation of element iodine [I-][I-]00+[IO3-]+[IO3-]00=[I-]+[IO3-]=[I-]+[IO3-] a=[IO3-], b=[I-]a=[IO3-], b=[I-] a0+b0=a+b a0+b0=a+b da/dt=kda/dt=k00(a0-a)-(a0-a)-kkcca(a0+b0-a)a(a0+b0-a)22

steady state da/dt=0steady state da/dt=0 a=aa=ass ss

kk00(a-a(a-assss)-k)-kccaassss(a0+b0-a)(a0+b0-a)22=0 one or three solution =0 one or three solution kk00(a-a(a-assss)=k)=kccaassss(a0+b0-a)(a0+b0-a)2 2 aass ss 与 b0 b0 和和 k0k0 有关有关

Page 7: Chapter 5        Bistability and oscillations              in flow reactors

dimensionless equationdimensionless equation ααssss=a=assss/ao β/ao β00=b0/a0 К=b0/a0 К00=k0/kc=k0/kca0a0

22

КК00(1-α(1-αssss)= α)= αss ss (1+β(1+β00-α-αssss))22

F RF R F=R steady state αF=R steady state αssss when βwhen β00=0.2 =0.2 1-α1-αssss=0 reaction begin , R: curve parabola F: line =0 reaction begin , R: curve parabola F: line

only one intersectiononly one intersection When βWhen β00=0.05 three intersections two are stable =0.05 three intersections two are stable States are decided by flowrate, [I]States are decided by flowrate, [I]0 0 and history and history

Page 8: Chapter 5        Bistability and oscillations              in flow reactors

5.4 Turning points and tangancies5.4 Turning points and tangancies

Line2,4 are tangential to RLine2,4 are tangential to R ,, The system is bistable between line2 The system is bistable between line2 and line 4and line 4

The condition for tangancyThe condition for tangancy

F=R dF/dF=R dF/dαα=dR/d=dR/dαα

The stable concentration of The stable concentration of α αα αssss±=0.25{3±(1-8β0)0.5}±=0.25{3±(1-8β0)0.5}

To makeαTo makeαssss real, β0<1/8 real, β0<1/8The flowrate of points of tangancies( Turning points)The flowrate of points of tangancies( Turning points)

Phase diagramPhase diagram

Page 9: Chapter 5        Bistability and oscillations              in flow reactors

5.5 Nodes and saddles: from bistable states to saddle-node 5.5 Nodes and saddles: from bistable states to saddle-node bifurcationbifurcation

In bistable region, there are three steady states, (or three branchs), In bistable region, there are three steady states, (or three branchs), two are stable, the middle is unstable, How to understand it?two are stable, the middle is unstable, How to understand it?

For their stability, We use the potential For their stability, We use the potential rate=-dα/dt= α(1+β0-α)rate=-dα/dt= α(1+β0-α)22-К0(1-α)-К0(1-α) V=1/2 m RateV=1/2 m Rate22 dV/drate=RatedV/drate=Rate V=∫Rate d(Rate) V=∫Rate d(Rate)

V=0.5(1+β0)2α2-2/3(1+β0)α3+1/4α4-V=0.5(1+β0)2α2-2/3(1+β0)α3+1/4α4-КК00α(1-1/2α)+V0α(1-1/2α)+V0At one parameter, Vα1, α3 has the minimum as node, and Vα2 hasAt one parameter, Vα1, α3 has the minimum as node, and Vα2 has the maxmum as saddle. when the parameter changes to another,the maxmum as saddle. when the parameter changes to another,Vα3 merges with Vα2 from node to saddle. This process is called saddle-noVα3 merges with Vα2 from node to saddle. This process is called saddle-no

de bifurcation. de bifurcation.

Page 10: Chapter 5        Bistability and oscillations              in flow reactors

5.6 Designing oscillatory reactions from bistable syste5.6 Designing oscillatory reactions from bistable systemsms

A Nonlinear feedback reaction (quadratic and cubic) + CSTR: only bistabiA Nonlinear feedback reaction (quadratic and cubic) + CSTR: only bistability ? lity ?

Feedback---Clock (batch)------Bistabilty(CSTR)Feedback---Clock (batch)------Bistabilty(CSTR)Only a feedback can not bring out oscillations: a+b=2b Only a feedback can not bring out oscillations: a+b=2b Degree of Freedom =1 a fixed then b is fixed a+b=a0+b0Degree of Freedom =1 a fixed then b is fixed a+b=a0+b0

B B For oscillations, the system must have two degrees of freedomFor oscillations, the system must have two degrees of freedom a+b=2ba+b=2b b+c=pb+c=pC model analysis for CSTR oscillations C model analysis for CSTR oscillations

a+2ba+2b3b kc R13b kc R1 b+cb+cBC kBC k11 k k-1-1 R2 R2 independent variables two b c or bcindependent variables two b c or bc

Page 11: Chapter 5        Bistability and oscillations              in flow reactors

assumption: assumption: flowing in a bflowing in a b c and bc: no flowing or flowout kc and bc: no flowing or flowout k11, k, k-1-1 small small initial concentrations: ainitial concentrations: a00 b b00 c c00

Concentrations at specific time: a b c, bc=x a0+b0=a+b(kConcentrations at specific time: a b c, bc=x a0+b0=a+b(k11, k, k-1-1 s small )mall )

γ=x/a0 γ0=c0/a0 β=b/a0γ=x/a0 γ0=c0/a0 β=b/a0

Non negative feedback ( R2 un-included)Non negative feedback ( R2 un-included)

Negative includedNegative included

Page 12: Chapter 5        Bistability and oscillations              in flow reactors

Inflow rate can be auto-vary and cycledInflow rate can be auto-vary and cycled

B(β) big B(β) big b+cb+cBC BC BC (BC (γγ ) rise ) rise κκ0,eff0,eff increasesincreases system move automaticallyto rightsystem move automaticallyto right , make B drop at turning point ,and, make B drop at turning point ,and

b b ((ββ) ) +c+cBC BC ((γγ ) R2) R2 inducinginducing : : a+2b=3ba+2b=3b Jump to low branch. Jump to low branch. B small, R2 equilibrium to left, BC (B small, R2 equilibrium to left, BC (γγ ) drops, κ0,eff decreases, syst) drops, κ0,eff decreases, syst

em moves to left, then B increase to turning point, jump up. So oem moves to left, then B increase to turning point, jump up. So oscillations repeated.scillations repeated.

Page 13: Chapter 5        Bistability and oscillations              in flow reactors

C To understand dynamics from nullclinesC To understand dynamics from nullclines

β nullcline γ nullcline intersection situation have four possibilitiesβ nullcline γ nullcline intersection situation have four possibilities..

Bistable high Bistable high β low β oscillationsβ low β oscillations

Cross-shapes diagrams Parameter κ0 γ0Cross-shapes diagrams Parameter κ0 γ0 ‘‘

a bistable d oscillations b high β c lowβa bistable d oscillations b high β c lowβ near cusp perturbation oscllations or go to stable statenear cusp perturbation oscllations or go to stable state If b c display oscillations, then a is the field of birhythmicity and If b c display oscillations, then a is the field of birhythmicity and

d is the field of complex d is the field of complex oscillations or chaososcillations or chaos

Page 14: Chapter 5        Bistability and oscillations              in flow reactors

5.7 Applications of Cross-shaped Diagram Technique5.7 Applications of Cross-shaped Diagram Technique

Bistable states in CSTR + Negative feedback Bistable states in CSTR + Negative feedback Autocatalysis + autocatalyst consume,This make the k0,eff chAutocatalysis + autocatalyst consume,This make the k0,eff ch

ange from parameter to variable. Complex dynamics such as ange from parameter to variable. Complex dynamics such as oscillations take places. oscillations take places.

IO3--AsO3- iodide autocatlysisIO3--AsO3- iodide autocatlysis, CIO2- as negative substances, CIO2- as negative substances

IO3-+5I-+6H+=3I2+3H2OIO3-+5I-+6H+=3I2+3H2O

I2+ H2O+H3AsO3=2I-+H2AsO4-+2H+I2+ H2O+H3AsO3=2I-+H2AsO4-+2H+

d[I-]/dt=(ka1+Ka2[I-])[I-][IO3-][H+]2d[I-]/dt=(ka1+Ka2[I-])[I-][IO3-][H+]2

IO3--AsO3-—ClO2- oscillationsIO3--AsO3-—ClO2- oscillations

Landolt reaction + Fe(CN)Landolt reaction + Fe(CN)663- 3-

V Gaspar and k. Showalter JPC, 94, 4973V Gaspar and k. Showalter JPC, 94, 4973

Page 15: Chapter 5        Bistability and oscillations              in flow reactors

220℃0℃ 30℃ 40℃30℃ 40℃

Page 16: Chapter 5        Bistability and oscillations              in flow reactors
Page 17: Chapter 5        Bistability and oscillations              in flow reactors

5.8 Complex oscillations and Chaos5.8 Complex oscillations and Chaos

More than two variables, The system can display complex More than two variables, The system can display complex oscillations, chaos Bifurcation to chaososcillations, chaos Bifurcation to chaos

Period-doubling quansiperiod Period-doubling quansiperiod

Page 18: Chapter 5        Bistability and oscillations              in flow reactors

Mixed-Mode oscillationsMixed-Mode oscillations

Page 19: Chapter 5        Bistability and oscillations              in flow reactors

5.9 mushroom and isola5.9 mushroom and isola

Page 20: Chapter 5        Bistability and oscillations              in flow reactors

model explainationmodel explaination A+2B=3B Rate=kcab2A+2B=3B Rate=kcab2 B=C rate=ktbB=C rate=ktb a+b+c=a0+b0a+b+c=a0+b0

Steady state conditionSteady state condition

P= (k0+kt)P= (k0+kt)22/k0 dynamical flowrate/k0 dynamical flowrate

Page 21: Chapter 5        Bistability and oscillations              in flow reactors

P= (k0+kt)P= (k0+kt)22/k0 /k0

Situation1 : k0>>kt P→k0Situation1 : k0>>kt P→k0

Situation2 : k0→0 P→∞Situation2 : k0→0 P→∞

Situation3 : Pmin=4kt Situation3 : Pmin=4kt Monostability if PMonostability if Pminmin> F> Ftangancytangancy