1. 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 If the reactor is well stirred, we call it CSTR(continuous-flow stirred tank reactor) Molecules can spend different time at the CSTR because of flowing. The average time spent in reactor is called the mean residence time T res = volume/ flow rate. The system can display steady state, bistable state, oscillations, chaos in a CSTR. http://www.grc.org/programs.aspx?year=2 008&program=oscillat reactants stirrer

2. 5.1 Steady state and bistability Example: Iodate - arsenite system IO 3 - +5I - +3H3AsO 3 → 6 I - +3 H3AsO 4 IO 3 - +5I - +3H3AsO 3 → 6 I - +3 H3AsO 4 At high flow rate, the concentrations of reactants in a CSTR have not bigger different with the inflow concentrations, this branch is known as flow branch( 流动分枝 ). At high flow rate, the concentrations of reactants in a CSTR have not bigger different with the inflow concentrations, this branch is known as flow branch( 流动分枝 ). At low flow rate, the system in a CSTR approach the thermodynamic equilibrium, the curve is called thermodynamic branch( 热力学分枝 ) At low flow rate, the system in a CSTR approach the thermodynamic equilibrium, the curve is called thermodynamic branch( 热力学分枝 ) As the flowrate is decreased, the system changes from flow branch to thermodynamic branch Attention: when flowrate change, the system evolve to new state for short(low flowrate) or long (high flow rate)time. Attention: when flowrate change, the system evolve to new state for short(low flowrate) or long (high flow rate)time. reacta nts stirrer

3. Situations: Situations: 1. When changing flowrate up or down, the plot of steady concentrations have only one curve. But during the some region of flowrate, there is a quick change of concentration from one branch to another branch. flow diagram

4. 2. Thermodynamic branch and flow branch overlap as the direction of changing flowrate is opposite. This is phenomena is named bistabilty. Within the region of bistability, the actual state selected depends on not only the parameter but also the operational history. When the flowrate is up or down, the state jumps or fall off in delay. This phenomena is called hysteresis When the flowrate is up or down, the state jumps or fall off in delay. This phenomena is called hysteresis This abrupt changes in compositions is discontinuous responded to the continuous changes in the operating conditions. This is bifurcation. This abrupt changes in compositions is discontinuous responded to the continuous changes in the operating conditions. This is bifurcation.

5. 5.2 Dynamic equations for flow reactors The changes of concentrations in flow reactor result from the net flow and reaction The changes of concentrations in flow reactor result from the net flow and reaction V dA/dt=q(A 0 -A)+VR dA/dt=k0(A0-A)+R k 0 =q/V time -1 For iodate-arsenite system IO3-+5I-+6H+=6I-+3H2O R=(K a1 +K a2 [I-])[I-][ IO 3 - ][H+] 2 Mixed autocalysis R=(K a1 +K a2 [I-])[I-][ IO 3 - ][H+] 2 Mixed autocalysis d[IO 3 -]/dt=k0([[IO 3 -] 0 -[IO 3 -])- (Ka1+Ka2[I-])[I-][ IO3-][H+] 2

6. 5.3 Steady state solutions: flow diagrams 5.3 Steady state solutions: flow diagrams Considering cubic autocatalysis only in iodate-arsenite system d[IO 3 - ]/dt=k 0 ([[IO 3 -] 0 -[IO 3 -])- (Ka1+Ka2[I-])[I-][ IO3-][H+] 2 Ka1=0 Ka1=0 k c =ka2[H + ]2 k c =ka2[H + ]2 d[IO 3 -]/dt=k0([[IO 3 - ] 0 -[IO 3 - ])- k c [I - ] 2 [ IO3-] conservation of element iodine conservation of element iodine [I-] 0 +[IO3-] 0 =[I-]+[IO3-] [I-] 0 +[IO3-] 0 =[I-]+[IO3-] a=[IO3-], b=[I-] a=[IO3-], b=[I-] a0+b0=a+b a0+b0=a+b da/dt=k 0 (a0-a)-k c a(a0+b0-a) 2 da/dt=k 0 (a0-a)-k c a(a0+b0-a) 2 steady state da/dt=0 steady state da/dt=0 a=a ss a=a ss k 0 (a-a ss )-k c a ss (a0+b0-a) 2 =0 one or three solution k 0 (a-a ss )-k c a ss (a0+b0-a) 2 =0 one or three solution k 0 (a-a ss )=k c a ss (a0+b0-a) 2 a ss b0 和 k0 有关 k 0 (a-a ss )=k c a ss (a0+b0-a) 2 a ss 与 b0 和 k0 有关

7. dimensionless equation α ss =a ss /ao β 0 =b0/a0 К 0 =k0/kc a0 2 α ss =a ss /ao β 0 =b0/a0 К 0 =k0/kc a0 2 К 0 (1-α ss )= α ss (1+β 0 -α ss ) 2 К 0 (1-α ss )= α ss (1+β 0 -α ss ) 2 F R F R F=R steady state α ss F=R steady state α ss when β 0 =0.2 when β 0 =0.2 1-α ss =0 reaction begin, R: curve parabola F: line only one intersection only one intersection When β 0 =0.05 three intersections two are stable States are decided by flowrate, [I] 0 and history States are decided by flowrate, [I] 0 and history

8. 5.4 Turning points and tangancies Line2,4 are tangential to R ， The system is bistable between line2 and line 4 Line2,4 are tangential to R ， The system is bistable between line2 and line 4 The condition for tangancy F=R dF/d α =dR/d α F=R dF/d α =dR/d α The stable concentration of α α ss ±=0.25{3±(1-8β0)0.5} To makeα ss real, β0<1/8 The flowrate of points of tangancies( Turning points) Phase diagram Phase diagram

9. 5.5 Nodes and saddles: from bistable states to saddle-node bifurcation In bistable region, there are three steady states, (or three branchs), two are stable, the middle is unstable, How to understand it? For their stability, We use the potential rate=-dα/dt= α(1+β0-α) 2 -К0(1-α) rate=-dα/dt= α(1+β0-α) 2 -К0(1-α) V=1/2 m Rate 2 V=1/2 m Rate 2 dV/drate=Rate dV/drate=Rate V=∫Rate d(Rate) V=∫Rate d(Rate) V=0.5(1+β0)2α2-2/3(1+β0)α3+1/4α4- К 0 α(1-1/2α)+V0 At 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-node bifurcation.

10. 5.6 Designing oscillatory reactions from bistable systems A Nonlinear feedback reaction (quadratic and cubic) + CSTR: only bistability ? Feedback---Clock (batch)------Bistabilty(CSTR) Only a feedback can not bring out oscillations: a+b=2b Degree of Freedom =1 a fixed then b is fixed a+b=a0+b0 Degree of Freedom =1 a fixed then b is fixed a+b=a0+b0 B For oscillations, the system must have two degrees of freedom a+b=2b a+b=2b b+c=p b+c=p C model analysis for CSTR oscillations a+2b  3b kc R1 a+2b  3b kc R1 b+c  BC k 1 k -1 R2 b+c  BC k 1 k -1 R2 independent variables two b c or bc independent variables two b c or bc

11. assumption: assumption: flowing in a b flowing in a b c and bc: no flowing or flowout k 1, k -1 small c and bc: no flowing or flowout k 1, k -1 small initial concentrations: a 0 b 0 c 0 initial concentrations: a 0 b 0 c 0 Concentrations at specific time: a b c, bc=x a0+b0=a+b(k 1, k -1 small ) Concentrations at specific time: a b c, bc=x a0+b0=a+b(k 1, k -1 small ) γ=x/a0 γ0=c0/a0 β=b/a0 γ=x/a0 γ0=c0/a0 β=b/a0 Non negative feedback ( R2 un-included) Non negative feedback ( R2 un-included) Negative included Negative included

12. Inflow rate can be auto-vary and cycled B(β) big b+c  BC BC (γ) rise B(β) big b+c  BC BC (γ) rise  κ 0,eff increases  κ 0,eff increases system move automaticallyto right system move automaticallyto right, make B drop at turning point,and, make B drop at turning point,and b (β) +c  BC (γ) R2 b (β) +c  BC (γ) R2 inducing ： a+2b=3b inducing ： a+2b=3b Jump to low branch. Jump to low branch. B small, R2 equilibrium to left, BC (γ) drops, κ0,eff decreases, system moves to left, then B increase to turning point, jump up. So oscillations repeated. B small, R2 equilibrium to left, BC (γ) drops, κ0,eff decreases, system moves to left, then B increase to turning point, jump up. So oscillations repeated.

13. C To understand dynamics from nullclines β nullcline γ nullcline intersection situation have four possibilities. β nullcline γ nullcline intersection situation have four possibilities. Bistable high β low β oscillations Bistable high β low β oscillations Cross-shapes diagrams Parameter κ0 γ0 ‘ Cross-shapes diagrams Parameter κ0 γ0 ‘ a bistable d oscillations b high β c lowβ a bistable d oscillations b high β c lowβ near cusp perturbation oscllations or go to stable state near cusp perturbation oscllations or go to stable state If b c display oscillations, then a is the field of birhythmicity and d is the field of complex If b c display oscillations, then a is the field of birhythmicity and d is the field of complex oscillations or chaos oscillations or chaos

14. 5.7 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 change from parameter to variable. Complex dynamics such as oscillations take places. Autocatalysis + autocatalyst consume,This make the k0,eff change from parameter to variable. Complex dynamics such as oscillations take places. IO3--AsO3- iodide autocatlysis, CIO2- as negative substances IO3-+5I-+6H+=3I2+3H2O IO3-+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+]2 d[I-]/dt=(ka1+Ka2[I-])[I-][IO3-][H+]2 IO3--AsO3- — ClO2- oscillations IO3--AsO3- — ClO2- oscillations Landolt reaction + Fe(CN) 6 3- V Gaspar and k. Showalter JPC, 94, 4973

15. 2 0 ℃ 30 ℃ 40 ℃

17. 5.8 Complex oscillations and Chaos More than two variables, The system can display complex oscillations, chaos Bifurcation to chaos Period-doubling quansiperiod Period-doubling quansiperiod

18. Mixed-Mode oscillations

19. 5.9 mushroom and isola

20. model explaination model explaination A+2B=3B Rate=kcab2 A+2B=3B Rate=kcab2 B=C rate=ktb B=C rate=ktb a+b+c=a0+b0 a+b+c=a0+b0 Steady state condition P= (k0+kt) 2 /k0 dynamical flowrate

21. P= (k0+kt) 2 /k0 Situation1 : k0>>kt P → k0 Situation1 : k0>>kt P → k0 Situation2 : k0 → 0 P → ∞ Situation2 : k0 → 0 P → ∞ Situation3 : Pmin=4kt Situation3 : Pmin=4kt Monostability if P min > F tangancy