Oscillating Reactions Web Module

In this section we will carry out a linearized stability analysis of the reactions just discussed to determine the start and end time of the oscillations, the frequency, and the number of oscillations. Even though an abbreviated form of the BZ reaction is given on the web below, it is still relatively complex. Consequently to illustrate how oscillating reactions occur we shall use the following relatively simple reaction sequences. Consider the following elementary multiple reactions. 

(1)                                                

(2)                                                             

(3)                                                   

(4)                                                             

Batch mole balances and combined rate laws

The initial conditions are that when t = 0 then: C_P = C_P0 and C_A0 = C_B0 = 0

1. Applying the PSSH

            Concentration

            Solving for the pseudo-steady state concentrations

We will now put the equations in Dimensionless form by

then

At Pseudo-steady state

            Dimensionless

At t_cross

            (27)        

At t_max

            (28)        

                            reaches a maximum at

2. Dynamics of the Reactions (Graduate Course Work)  

    2.A. Linearizing the equations to obtain an analytical solution

Recall the dimensionless form of the combined mole balances and rate law is:

            (29)      

            (30)      

Let Da and Db be perturbations from the pseudo-steady state.

We will now substitute for a and b and expand f(a,b) and g(a,b) in a Taylor series around the Pseudo Steady State values a_ss and b_ss.

By recalling f(a_ss,b_ss) and g(a_ss,b_ss) are zero  

Then,                    

Where,       

For which the solution to Equations (32) and (33) is  

            (36)      

            (37)      

where the roots of the solution matrix are

            (40)        

See Properties of l's

Hopf Bifurcation

The real part of l₁, l₂ is zero, i.e. Tr(F) = 0

An exchange of stability from a stable critical point to a limit cycle, a bifurcation of this type, from an equilibrium to a periodic solution is called a Hopf bifurcation.

2.B. Finding the conditions that give oscillatory behavior

For oscillating behavior, Tr(F) = 0

recalling    

           (41)        

and

            (42)        

and

we now evaluate the expansion terms in the Taylor series

            (43)        

            (44)        

            (45)        

            (46)        

(2)                          

We will now evaluate the determinate

Det

            DET       

               , however, for PSSH, b_SS = m

Det (F) = m² + k_u / QED

We now want to consider the case where we have pure oscillations, Tr(F)=0

Setting we find

Solving for the roots which will make Tr=0, and produce pure oscillation

            (48)        

                           where m_1,2 are the roots of the equation with  taking the + sign

Find the start and end of oscillations 

The oscillations begin when m = m₁* and end when m = m₂*

We can now calculate the times at which the oscillations begin and end.

                           we see that we start at t=0 and m=m_o and that m decreases with t (time). The oscillations start at when m reaches . The corresponding dimensionless time is

                           The oscillations end when m reaches .

The figure below shows how the roots vary with k_u.

                           The dimensionless frequency of oscillation is

                           The figure below show the frequencies of oscillation near the start  and end  of the oscillations as a function of k_u.

        Finding the periods of the oscillation

                           The dimensionless period of the oscillation

                           At the start of the oscillations

                           Near the end of the oscillation

                           The figures below shows the dimensionless periods of oscillation,  and  as a function of k_u.

                           The geometric mean period of oscillation

                           The time of the oscillation is

                           The number of oscillations

                           Dimensionless mean frequency of oscillation

  1)   Reconsider the problem on oscillating reactions. A 10°C increase in temperature produced the following observations. The dimensionless times at which the oscillations began and ended decreased. The dimensionless period of the oscillation at the start of the oscillation increased while the dimensionless period near the end of the oscillation decreased. What conclusions can you draw about the reactions? Explain your reasoning. Feel free to use plots/sketches or equations if you wish.

2)   Oscillating Reactions

(1)   Under what conditions would the time at which oscillations begin (e.g. t = 1800s) increase as the temperature increases by 10°C?

(2)   Under what conditions would the time at which the oscillation begin and end remain the same?

(3)   Under what conditions would the period of oscillation in seconds at both the start and end of the oscillation time decrease with increasing temperature?

Adapted from Chemical oscillations and Instabilities, by Peter Gray and Stephen K. Scott, Oxford Science Publications, 1990.

Appendix 1

Two coupled ODEs

Click back (page 4 Oscillating Reactions)

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Appendix 2 

Determining the properties of the roots l₁, l₂

1.                  If Tr(F) < 0 ,  Det (F) > 0 ,  [Tr2(F) – 4 Det(F)] > 0

            Both l₁ and l₂ are real and negative

                        Critically Damped

2.                  Tr(F) < 0 ,  Det (F) > 0 ,  [Tr²(F) – 4 Det(F)] < 0

3.                  Tr(F) > 0 ,  Det (F) > 0 ,  [Tr²(F) – 4 Det(F)] < 0

            Unstable oscillations

4.                  Tr(F) > 0 ,  Det (F) > 0 ,  [Tr²(F) – 4 Det(F)] > 0

Both roots positive + and real

                                        unstable

5.                  Tr(F) = 0 ,  Det (F) > 0 ,

               ,

               ,

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