Computational Fluid Mechanics
Hydrodynamic Instabilities


Said Ghalimi (mfn06)





 The Computer Model






The behavior of oscillatory phenomena has become an active area of research in the past two decades, and many complex systems have been studied and modeled. The subject of this work, the soda bottle oscillator, is relatively simple and easy observed : shake a bottle of soda water and at temperature room several times, let it stand a minute, then puncture the cap with a thumb tack or similar object. As pressure is released from the bottle, the dissolved gas is seen to bubble of in oscillatory bursts rather than in a steady single flow.

In order for far-from-equilibrium systems to exhibit oscillatory behavior, the mechanism must contain a nonlinear step and a step in which feedback inhibits the nonlinear step. Oscillations in reactions that are for interest to chemists can have either a chemical or a physical mechanism. Reactions such as the Belousov-Zhabotinsky reaction exhibits chemical oscillations, whereas the oscillations in gas evolution processes, such as the Morgan reaction, have a physical origin.

Chemical oscillations are effect of reaction kinetics, and it is the concentration of intermediate chemical species that oscillates. To exhibit oscillatory behavior, a reaction mechanism must have an auto-catalytic step and a step that produces a species that inhibits the auto-catalysis. The inhibitory step provides the feedback that controls the nonlinear (auto-catalytic) step. The complete mechanism for an oscillating chemical reaction is complex, as are the models that illustrate their behavior.

In a gas evolution oscillator (GEO), a dissolved gas (carbon monoxide from formic acid in the Morgan reaction) is produced by chemical reaction at atmospheric pressure. Once the solution becomes fully supersaturated, homogeneous nucleation of bubbles occurs. Diffusion of dissolved gas into growing bubbles depletes the solution sufficiently to shut off further bubble formation, and this is the feedback responsible for oscillation. The oscillations are sustained by continuous chemical production of the dissolved gas.

The mechanism of oscillation in the soda bottle oscillator is also physical. However, it is very different from the mechanism in a GEO. We believe that the oscillations in this case occur as the system returns to equilibrium in response to the changing pressure of CO2 above the solution.

The process starts with dissolved gas in equilibrium with gas in the headspace, in accordance with Henry's law :


where ceq is the concentration of dissolved gas, p is the external pressure and k is the Henry's law constant. When a small hole is made in the cap, the initial decrease in pressure causes the onset of bubble nucleation. As the bubbles escape into the headspace, they tend to increase the pressure, and if the rate of increase is greater than rate of decease due to pinhole, then the net result is that the headspace becomes partially repressurized. This is the feedback that turns off the nucleation process. The cycle repeats several times before reaching equilibrium atmospheric pressure.

Models can provide new insights into systems. Based on the above mechanism, we describe here a computer model, "the bottelator", for oscillations in the soda bottle, and show that the model is in good accord with the experimentally measured oscillations. Although the bottelator is relatively simple, the nucleation of bubbles, which is central to our model, is neither simple nor well understood. Studying the behavior of models such as this may eventually contribute to our understanding of the difficult problem of phase nucleation.


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- Recording the oscillations -

- The computer model -

- Discussion -


Recording the oscillations :

A 1-L bottle of commercial Club Soda was shaken vigorously at room temperature and allowed to sit for about one minute. A small hole was made in the cap with a thumbtack, and the bottle was quickly connected to 5-L ballast vessel with wide pressure tubing. An omega PX-160 pressure transducer was attached to the ballast, with a capillary leak to the atmosphere. The transducer signal was recorded on a conventional strip-chart recorder. Figure-1 is a typical trace of the oscillations obtained. Judged visually, the maximum rate of bubble escape corresponded closely in time with the peak of the ballast pressure, so that the curve shown in Figure-1 is a fairly good indicator of the bottle headspace pressure.




The Computer model :

The oscillations where simulated using the Stella II modeling package (High Performance System, Hanover, NH). Figure-2 shows the flow diagram set up as the basis of the model. Stella II "reservoirs" are used to represent the dissolved gas in the solution and the gas pressure in the headspace. A "conveyer" is used to represent the formation, rise and escape of bubbles. Outflows determine the rate at which the contents of the reservoirs change. The outflow from the dissolved gas is set by the nucleation rate, and the escape of gas from the headspace is modeled as a first-order process. The escape of bubbles from the conveyor is represented by the conveyor outflow.

The design of the model was based on the qualitative explanation of the oscillations given above. However, several simplifying assumptions were made in the formulation. The first important simplification involves the use of the conveyor/conveyor outflow to represent the gas bubbles. The conveyor models a time delay between the formation of bubble nuclei and their escape into the headspace. A further simplification was the complete neglect of the growth of rising bubbles : it was assumed that bubbles of a uniform size either are formed by nucleation and escape after a defined period of time, or they do not exist. This simplifying assumption does not take into account that some of the small bubble nuclei that form will redissolve rather than escape. The point will be discussed later.

Outflow form the dissolved gas reservoir (that is, bubble nucleation) is defined by EQ-2 :


where J is the bubble nucleation rate, a is related to the rate of encounters of dissolved gas molecules, and k is Boatsman's constant. The change in Helmholtz energy to from a bubble, D A, can be shown from classical nucleation theory to be :


where s is the surface tension of the liquid and cs and ceq are the super-saturation and equilibrium concentration of dissolved gas, respectively. Equations 2 and 3 combine to give :


where b is a constant depending on s and k . Equation 4 shows a sharp, almost discontinuous rate in J at a critical value of (cs-ceq). It is this nonlinear characteristic of bubble nucleation that is necessary for oscillations. Parenthetically, EQ-4 also illustrates the essential difference between a GEO and the soda bottle oscillations. In the former, criticality is attained as cs increases by chemical reaction, whereas, in the latter it is reached when ceq decreases, in accordance with Henry's law, as the headspace pressure decreases.


 The Nucleation rate J function of Cs.

Two difficulties arise using EQ-4 in the model. The first is that it is known that employing the macroscopic surface tension to calculate b leads to very poor qualitative agreement with experiment for the nucleation concentration. The second problem is that bubble nucleation in the soda bottle is not truly homogeneous. For these reasons, a and b were designed as adjustable parameters in the modeling.

The equation for the escape of gas from the headspace involves a rate constraint, ke, multiplied by the gas pressure in the headspace. This constant was also used as an adjustable parameter.

dCs/dt = -J(Cs,Ceq)

dCb/dt = J(Cs,Ceq) - DEF.Cb

dCs/dt = DEF.Cb - Ke.Cs

The Stella II modeling package allows adjustable parameters to be represented as "sliders". The sliders make it possible to change parameters throughout a specified range and observe the effects while the model is running. In this way the parameter values that gave rise to oscillations were determined. The values used in these computer simulations were a =3.104; b =250; and ke=6. The effects of changing these parameters are discussed below.

These initial conditions used in the model were gas pressure in the headspace, 5.105Pa; dissolved gas, 170 mole m-3; and gas bubble in conveyor, 0. The program was run using fourth order Range-Kutta as the method for numerical integration, with a time step of 0.001.




Discussion :

Periodic onset and shut-off of bubble nucleation as a function of external pressure can be visualized as a trajectory on a phase diagram, as shown in Figure-4. There are two "solubility " lines in Figure-4: the Henry law defined by EQ-1, and the Super-Henry law line, which shows how the homogeneous nucleation limit (cnuc) changes with external pressure. The SHL equation is easily derived from equations (1-4): if cnuc0 is the limiting nucleation concentration at zero external pressure, we obtain, after some algebraic manipulation,


the pinhole is made at point B1 on the HL line where the system is in equilibrium at a headspace pressure of about 5 atm. As the gas pressure in the headspace decreases, the dissolved gas concentration stays constant until it reaches cnuc, where the trajectory intersects the SHL line at point X1. The trajectory then travels diagonally down to the right as homogeneous nucleation decreases the dissolved gas concentration and increases the gas pressure in the headspace. The nucleation process shuts off when the trajectory intersects the Henry's law (HL) line at point B2 and the cycle begins again. The trajectory will snake between the two lines until the headspace pressure reaches 1 atm. Any excess dissolved gas remaining at that point will escape in lagging bubbles or by diffusional loss across the surface.

The trajectory shown in Figure-4 is highly idealized because it assumes homogeneous nucleation. The actual formation of bubbles in the soda bottle oscillator is probably not predominantly by homogeneous nucleation. Miniscule and persistent cavitations are created when the soda bottle is shaken prior to puncture. These and the vessel walls act as centers for heterogeneous nucleation. The equation for heterogeneous bubble nucleation (5,6) is similar to equation 4 except that interpretation of the quantities a et b differs. Our aim was to demonstrate that a variation in nucleation rate with external pressure, using a nucleation type equation, does produce oscillations. A system in which the primary mechanism for nucleation is heterogeneous will show the same qualitative behavior as a system in which nucleation is homogeneous, but the degree of super-saturation attained will be less. The trajectory will travel horizontally only part of the way to point X1 on the SHL line before super-saturation is released heterogeneously.

For comparison, an idealized trajectory for a GEO is also shown in Figure-4. The path starts at point A (at 1atm pressure) and travels vertically upwards to point M, then oscillates along the line MN until the reaction is over.

In our original proposition of the mechanism for the soda bottle oscillator, we considered several factors : (i) bubble nucleation occurring as in equation 4; (ii) the time delay between nucleation of bubble and its release of gas into the headspace; (iii) redissolution of small bubbles; and (iv) the effect of growth of rising bubbles. Oscillations were still observed in this model when redissolution of small bubbles in the solution and the further depletion of dissolved gas by growing bubbles were neglected. Since these factors change the dissolved gas concentration, the actual external pressure at which bubble nucleation occurs and is shut off probably differs in the computer simulation and the real system, but this does not affect the overall behavior of the bottelator. We conclude that while process iii and iv undoubtedly occur, they are not essential for oscillatory behavior.

Although values of a , b and ke were adjustable, the model did demonstrate that only a narrow range of values for these parameters produces oscillations. Actual experiments with the soda bottle oscillator produce good oscillations in only about one-half of the bottles tried; that is, certain rather specific conditions give rise to oscillations. For example, if the escape rate of gas through the pinhole is too rapid (pinhole too large) the dissolved gas escapes in one burst. This is easily demonstrated with the computer model: a small increase in the parameter ke causes the model to blow up.



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 Concentrations (Cs, Cb, Cs)




 Concentration des bulles dans la bouteille.



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