Far from Equilibrium-The Continuous-Flow Bottle Leonard J. Soltzberg Simmons College, Boston, MA 021 15 The study of chemical and physical systems far from equilibrium shines bright with conceptual promise, as did wave mechanics in an earlier era. Research in this area has begun to flourish. and there a m e a r to he imnortant a ~ ~ l i c a t i oin ns chemistry,'physics, and kology. Our own schooling as chemists. of course. has focused on hehavior a t equilibrium in isolated or closed systems-reversible processes, phase equilibria. and so forth. In treating chemical reactions, we have generally considered clused systems i n which we expected reactant and product roncentrations to change in a smuorh, monotonic manner and intermediate concentrations to conform to the steady-state approximation. And we have certainly been trained to expect that the composition of a reaction mixture should be uniform in space. I t is now known that there are many real physical, chemical, and biological processes that display quite different behavior. Surnrisine"ohenomena occur in open systems that . are far from equilibrium. This novel hehavior, sometimes referred to as "self-organization", can be characterized as breaking expected symmetry in time andlor in space. For chemical processes, time symmetry-breaking gives rise to oscillating chemical reactions, and spatial symmetry-breaking corresponds t o departure from uniform reaction mixture, as with spiral and concentric chemical waves ( I ) . The basis for current understanding of such events comes from the science of nonlinear dynamics, an area unfamiliar to many chemists. We report here a simple model system which makes it possible to demonstrate the major concepts of nonlinear dynamics and to observe several common farfrom-equilibrium phenomena. While the model presented here is a physical model, its behavior is remarkably analogous to that of chemical systems.
regular oscillatory flows, a simple H20 U-tube manometer is also connected to the air space ahove the liquid in the flask. It is useful in observing certain flow patterns through the outlet to utilize an adjustable strobe light mounted near the nozzle. The "Dynamlcal" Descrlptlon A major consideration in the study of a far-from-equilibrium process is the method by which the process should be described. The basis for such description comes from the field of dynamics. As chemists have become interested in chemical reactions as "dynamical" systems, the language of dynamics has begun to appear with some frequency in the chemical literature. For a broad, qualitative introduction to dynamics, the reader might consult the series by Abraham ideas are also examined in a chemical and Shaw (3). . ~ These . context by Kpstein ( 4 ) . In what follows, we shall descril~ethe behmiorof the continuutm-flow bottle from thr viewpoint of dynamics and point out the several analogies withthe behavior of far-from-eauilibrium chemical svstems. \\'hen we perform exprriments on a system, we control the wlues of certain variat~les(called "constrniuffi" or "control variables") and observe the resulting hehavior of others (called "responses"). In demonstration experiments with the continuous-flow hottle, one constraint is the inlet flow rate, controlled by the water faucet. I t is easy to measure the flow rate. When the bottle is in a steady state (liquid level ~~~
~~~
The continuous-FIOWBottle Field (2) has suggested the familiar flow of heer from a hottle as a model for far-from-equilibrium behavior. Speaking of self-organization under the influence of an increasingly large disequilibrium, he says This [tendency toward complex behavior] is analogous to beer flowing out of a bottle, a process governed by highly nonlinear hydrodynamic laws. A bottle sitting upright on a tahle is at equilibrium. If it is lifted off the tableand tilted slightly, it will still he near equilibrium. Beer will flow out of the bottle in a smooth, uninteresting way because the driving force is small. This is the linear regime. Removing the hottle farther from equilibrium by laying it on its side may cause a gurgling (oscillatory)flow of beer. Finally, if the hottle is turned upsidedown the drivingforceis at a maximum, and beer will flow out with some oscillation hut also with elaborate soatial oreanization reflected hv the amearance of .. " rurhulenre. Fur this complru dynnmw t,chavior 10 p~rsist,the system would havt to be open-ie., the iwer would haw to be replaced as it poured out ot che bottle. The continuous-flow hottle described here is the open system version of the hottle suggested by Field (Fig. 1).it is constructed from a 500-mL three-neck standard taper flask. A replaceable nozzle allows water to flow out, whiie a compensating inlet flow is provided by a large-diameter hose attached to a water faucet. The air space ahove the liquid, important in the flow behavior, is provided with a capillary leak to the atmosphere. In order to facilitate observation of
Figure 1. Continucus-flow bottle made from 500-mL three-neck flask. The hole for the stopper. which connects the capillary leak and manometer. is made by heating asmall area of the flask with aglassblowing torch and blowing out the flask wall. The measured values h, xo. x. and 0 are needed to calculate the speed of the exit flow. We have found 60' to be a convenient nozzle angle.
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February 1987
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inside the flask neither rising nor falling), the outlet flow rate must equal the inlet flow rate. Thus, we collect water from the outlet in agraduated cylinder, timing thecollection period with a stopwatch; the flow rate is the volume divided by the time. The flow rate is a continuous variable, varying from 0 mL1s to about 150 mLls in our apparatus. The other constraint in this system is the leak the atmosphere. Although the leak operates by affecting the pressure in the space above the liquid, for simplicity we shall treat the leak constraint as a discrete variable with just two values-closed and open. The response variable that we measure in the continuousflow bottle is the speed of the water leaving the nozzle. This quantity is not the same as the exit flow rate (in milliters per second) hut is rather the averaee " soeed . (in centimeters ner second) of an imaginary small disk of water as it leaves the nozzle. We can measure the meed hv la cine a section of meter stick in line with the outlkt streamin thebottom of the sink and recording the point of impact x of the outlet stream; for calculating the speed, we also need the height h of the nozzle above the meter stick, the zero point xo on the meter stick directly below the nozzle, the angie B of the nozzle from the vertical (see Fig. I), and the gravitational accelerationg. . The speed is calcuiated as' speedZ= Ig(x - ~ , ) ~ /sinz 2 8lllh - [(x - x,)/tanO]l Alternative response variables, which we have not measured, would be the height of the liquid in the flow bottle and the pressure above the liquid. An experiment on this system, then, consists of setting the values of the constraints (inlet flow rate and leak) and recording the resulting response (speed). One might expect these experiments to he exceedingly dull. Blstabllity Surprisingly, the behavior of this system is far from dull. One surprise is that, foragivensetting of the constraints, we can observe two different values of the response. Suppose, with the leak closed, we turn on the faucet to a low initial flow rate. The water rises in the flask until i t reaches the level of the outlet. Water begins to trickle out the nozzle, and, when the outlet flow rate equals the inlet rate, the system reaches a steady state with a small stream of water running from the nozzle. In this flow mode, the water stream occunies the bottom troueh of the nozzle. and the upper part i; filled with a column~,fair, which ruinerrs the insideof the flilsk with thratmosnhere. Weshall refer to this flow mode as "Mode 2" (see ~ i g 2a). . As we gradually increase the flow rate, the steady-state height of the liquid in the flask rises, and the speed of the outlet stream increases, hut the flow is still Mode 2. Eventually, the level of the liquid inside the flask rises high enough to block the passage of air through the nozzle. A t this point, a few air bubbles are carried out of the nozzle by the water flow, the pressure
' To derive this relationship, we use Newton's Second Law of Motion and the geometry of the apparatus (Fig. 1). and we assume that air resistance does not affect the trajectory of the stream. If we call the speed of the stream exiting the nozzle vo, then the vertical distance the stream falls in time t is gt2/2 vot cos 0. When the stream has fallen the distance h, we have
+
in the same period of time t, the stream will have moved horizontally a distance
Eliminating t between these two equations and solving for vo2, we obtain the desired equation for the speed of the stream. 148
Journal of Chemical Education
Figure 2. Types of flaw through the nozzle. (a) "Mode 2" flow. (b) "Mode 1" flow. (c) Slow oscillation. (d) Rapid oscillation.
inside the flask drops below atmospheric pressure, and the flow switches to "Mode 1"; that is, the nozzle is entirely filled with water (seeFig. 2h). Let us call the flow rate at which this switch occurs C2. The speed of the outlet flow in Mode 1is considerably slower than in Mode 2, as evidenced by the fact that the impact point on the meter stick is much closer to the nozzle. The reason for this change in speed is that, to produce a given uolume flow per unit time, a lower speed is required if the entire cross section of the nozzle is filled with water. If we further increase the inlet flow rate, the speed of the outlet flow increases but remains well below the higher Mode 2 soeeds. Now, we begin to decrease the flow rate. The speed of the outlet flow begins to decrease. As we continue decreasing the flow rate, we &cover that the flow continues in Mode 1even a t flow rates well below the C pswitchover point. Finally, a t quite a low flow rate, the flow switches hack to Mode 2, with both water and air occupying the nozzle; let us call the flow rate at which this switch occurs Cj. The result< 01 this experiment can be summarized in a "nmstraint-resoonse" d o t , u ~ i n bhitein's s. terminolow (41. We plot the responseLparameter (speed) versus the-constraint parameter (inlet flow rate) (Fig. 3). The hysteresis apparent in this plot shows that, for a wide range of flow rates, the system can be in either of two different states (Mode 1 or Mode 2). Such behavior is called "histability" and is found in a number of chemical systems in open contin-
A constraint-response plot for behavior with the open leak is shown in Figure 4. Relationship between Blstabillty and Oscillation Boissonade and De Kepper (5, 6 ) have developed a con-
ceptual scheme that relates oscillation to bistability. This powerful model explains a t least the qualitative features of manv,~ chemical svstems and aives ouantitative agreement with somr Thc wntinuous-flmv huttle conforms to the Roisstlnade.De Keooer .. morlcl and thus ran hr used to illusrratr the principles involved in the model. The Boissonade-De KeDper model applies to Systems which have an intrinsic bisiability, such & t h e continuousflow bottle. It further requires that there he two constraints operating on the system with the property that one constraint implicitly affects the value of the other one. We can understanil this relationship in terms of the f l ~ wrate and Irnk cnnitraints that operate on the n,ntinuous-flow hottle. A s~eady-statr liquid le\.el inside the hottle isarhieved when the uurlrt !low rate equalsthe inlet flow rate. In Mode 2 flow, the outlet flow r3te is determined by the hydroctatic head of water (the height of the liquid level a h v e the lowrr edge of the enrranrr to the nozzlel. It w e increase the inlet flou rate, the water level inside theflask rises, thereby increasing the outlet flow rate until i t eauals the new inlet flow and producesa new, higher steady&ate level. In this phase of operation, the air space inside the bottle is always a t atmospheric presmre, Po, ;ia the air space in the nozzle; this means that, even if the leak a t the top of the flask is closed, there is an effective leak via the nozzle. But now, when the liquid level inside the flask rises high enough to obstruct the free passage of air through the nozzle, a few air bubbles are carried out the nozzle, and the pressure inside the flask drops perhaps 10 torr below atmos~hericpressure. This pressure drop slows down the outlet flow rate, and the liquid level inside the flask rises hiaher until the head is high enough to produce steady-state flow; this high head i s accompanied, of course, by Mode 1flow. We can summarize this situation by saying that the "effective inlet flow rate" (Cell),as evidenced by the high head and Mode 1flow, is higher than the actual inlet flow rate (Co),and that this departure (c) is due to the feedback between the flow rate and the leak. This relationship can be seen concretely by venting the opening at the top of the flask and watching the water level inside the flask dron back down to the lower steadv-state level correspondingio the actual inlet flow rate Co. According to Boissonade and De Kepper, i t is this interaction in which one constraint influences the effective value of the other that leads from bistability to oscillation. For this scheme to work, the interaction between the two constraints must be different in the two different states of the system. This characteristic is particularly marked in the continuousflow bottle: in Mode 1.the closed leak leads to an effective ~~~~
~
Figure 3. Constraint-response plot for continuous-flowbonle. Nozzle long. 15-mm i.d Nozzle angle 60°. Leak closed.
15 cm
flow tale, mllsec Figure 4. Constraint-response plot for continuous-flow bottle. Same nozzle. same angle. Leak open (20 cm of 0.5-mm capillary tubing).
uous-flow stirred-tank reactor studies (4, 5). In a bistable system, there are two different stable states for a given value of the control variable. Which state is realized depends on the initial conditions; in the continuous-flow bottle, the initial conditions are determined by whether we were increasing the flow in Mode 2 or decreasing the flow in Mode 1. Oscillation
If we open the leak so that the air space above the liquid is vented through a capillary to the atmosphere, the behavior changes dramatically. Starting at a low inlet flow rate, we observe Mode 2 high-speed flow as in the previous experiment. As the flow rate is increased, however, we arrive a t a critical point at which the flow begins to oscillate. That is, the speed of the outlet stream oscillates hack and forth between fast and slow speed, as evidenced by the swinging back and forth of the impact point of the water on the meter stick. Looking at the nozzle, we can see that the mode of flow is oscillating between Mode 2 and Mode 1 (Fig. 2c). The oscillation can also he observed conveniently by watching the level in the water manometer attached to the flask. As the flow rate is further increased, the period of the oscillation tends to hecome shorter, and the amplitude (the distance between the impact points on the meter stick) hecomes smaller. When the period of the oscillation becomes shorter than the transit time of a portion of water through the nozzle, the water in the faster-moving Mode 2 flow regions in the nozzle piles up into the slower-moving Mode 1 regions, and we observe a regular train of bubbles (empty spaces) moving down the nozzle (Fig. 2d); these can he ohserved to advantage hv illuminating the nozzle with the n gthe nozzle adjustable frequency strobe lamp. ~ e ~ e n d i on lenath and diameter, careful further increase of the flow rate mn produrea phrnwnenon nnalrgous to the"l)ursting"seen in chemical systems (51; the system switches hvtu,een s i n ~ o r hMode 1 t l w and the osrillati~ryhubhlr stream.' Increasing the flow rate yet further eventually raises the liquid lwei to the point that onlv Mode 1 flow issecn.
Somewhat differentflask configurations are optimal for demonstration of differentphenomena. A wide nozzle (15-mm i.d.) about 15 cm long is good for observing bistability and, in conjunction with a leak made from a 20-cm length of 0.5-mm capillary tube, gives slow oscillations that permit measurement of the outlet speed during oscillation. A narrower nozzle (8-mm i.d.)about 15 cm iong with the same leak gives faster oscillations, which can be observed with the manometer, and at the higher frequencyoscillations gives bubble trains in the nozzle that can be observed with strobe illumination. With the same leak, a yet narrower nozzle (7-mm i.d.)about 15 cm iong is good for observing bursting; the flow rate should be adjusted just below that wh ch produces pure Mode 1 flow n any conftguratnon.the period of osclllat'on can be mcreased by lncreasmq the lenqtn of the cap. l ary eak A short 6-cm nozzle of 7mm i.d. Ggether kith a wide-open leak (capillary removed) gives sharp transitions from pure Mode 1 flow to bubble trains to rapid oscillation between discrete Mode 1 and Mode 2 flows to pure Mode 2 flow a5 the inlet flow rate is decreased. ~
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February 1987
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3
slable, state 2
u
n
F
$
constmi-. constraint
,
state 2
slate 1 constraint
Figure 5. A folded manifold. (a)Threedimensional phase-space. (b) Projection onto two-variable surlace.
flow rate which is higher than the actual flow rate, while in Mode 2, the closed leak has no effect on the effective flow rate, since the flask is vented to the atmosphere through the nozzle. To appreciate this scheme, we turn t o a theoretical construct from the field of dynamics. The state of a system a t any particular time can be thought of as a point in a space whose coordinate axes are the variables which characterize the svstem: the "nhase soace" we utilize in statistical thermod;nami& is an example of such a space. For dynamical systems, the identification of the appropriate phase space for a system is a matter of some importance, and the phase space used to describe a system is called a "manifold". Com-
monly encountered manifolds are cylinders and toruses (3). Bistable and oscillating chemical systems can he described in terms of theMfoldedslow manifold" (5, 7) (see Fig. 5a). This surface represents a portion of the phase space for the svstem in ouestion: the curved "orecioices" at the edees of the fold a; the places where sharp tiansitions occur-from one state to another. If we select iust two variables for studv. .. we can pn~jecrthis manifold onto a plane, as in Figure 5h. Figure 6 hhows the r)miertion of the outative folded manifold for a typical systkm:~hereis a significant resemblance between this proiection and the constraint-resvonse hvsteresis loop associated with bistability (for example, the bistability of the continuous-flow bottle, Fia. 3). For the continuous-flow bottle, the inlet flow rate corresponds to the constraint C, and the outlet speed corresponds to the response variable. With the leak closed, C1 would be the critical flow rate for the flow transition Mode l-to-Mode 2, and Cz would be the critical noint for the reverse transition. We now consider the effect of the other constraint; for the continuousflow bottle, this is the leak. In the aeneral case described bv Hoissonade and De Kepper, the se&d variable changes the effective valueof the first variable, and theeffect isdifferent in the two different states. Thus, depending on the value of the second variable (the leak), the effective flow rate is not the actual value Co but Ca - ej, where ri represents the perturbations c1 (positive) in Mode 1 and t2 (negative) in Mode 2. In addition, the model requires that the response of C to the other constraint be slower than the response of the svstem state (resoonse variable) to chanees in C. Imaeine n k that the system is in Mode 2 at a flow'iate Co withiAhe histabilits range (Fig. 6).Due to the influence of the second constraini, theperturbation, 62 is such that the effective flow rate is not Co but rather a value C'.ff which lies above C 2 within the Mode 1 flow region; because of the slower response time of the coupling between constraints, the effective flow rate approaches C'.ff gradually, and the system will switch rapidly to Mode 1when the effective flow rate reaches CZ.But now in Mode 1, due to the different influence of the second constraint in Mode 1, the perturbation el characteristic of Mode 1beeins to disolace the flow rate from Cn to a value c",c~, whiccis below and in the flow region for Mode 2: as the effective flow rate eraduallv C".,G. . aooroaches .. .... the system will switch quickly back to Mode 2 when the effective flow rate reachcs C , . and the osrillatorv rvcle will reoeat. (Bear in mind that 21the while the act& flow rate is &.) According to this model, the conditions under which oscillation will occur are
t1
Co - r , < C, c, - C
(1)
(2)
Boissonade and De Kepper point out that we can alternatively view the effect of the second constraint as perturbing the values of the critical points C1 and Cz to new values C1' = CI €1 and Cz' = Cz cz. This is apparent if we recast eqs 1 and 2:
+
+
Co
