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J . Phys. Chem. 1991, 95, 1306-1 3 1 1
close to the critical Rayleigh number of 67.9 and an axisymmetric shape for larger Rayleigh numbers, as was observed. (3) There is no critical radius in the horizontal orientation, and convection always incrcascs thc wavc front velocity. Using the expression for thc asymptotic flow velocity of a differentially heated horizontal cylinder. wc can predict thc order of magnitude of the increase in front velocity caused by convection. I f a wave reaction were endothermic and had a negative reaction volume, then thc upward front would be the pure reaction-diffusion onc, and thc descending front would be affected by convection. The horizontal configuration would show the same behavior a b thc iodatc-arscnous acid system with the angle of the front rcvcrscd.
If an exothermic wave reaction has a negative reaction volume, or an endothermic reaction has a positive reaction volume, then a completely different mechanism will affect the front velocity. This scenario requires double-diffusive convection to explain the increase in front velocity, a problem that will be treated in part 3 of this series. Acknowledgment. This work was supported by the National Science Foundation (Grants CHE-8800169 to I.R.E. and CHE8920664 to K.S.). Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this research. We also thank Simeen Sattar for a critical reading of the manuscript.
Convective Effects on Chemical Waves. 3. Multicomponent Convection in the Iron( 11)-Nitric Acid System John A. Pojman,+ Istvin P. Nagy,i and Irving R. Epstein* Department of Chemistry, Brandeis University, Waltham, Massachusetts 02254-91 10- and Institute of Physical Chemistry. Kossirth Lajos University, H-4010 Debrecen, Hungary (Received: July 18, 1990)
We consider the effects of convection on chemical waves in which density gradients result from the exothermicity as well from the isothermal volume change of the reaction. The nitric acid oxidation of Fe(l1) supports a constant-velocity traveling front which can be followed by observing the brown ring of the intermediate FeNO*+. The front velocity is extremely sensitive to the width of the tube in which the reaction is performed as well as to the orientation with respect to the gravitational vector. Aaccnding fronts propagate faster than pure reaction-diffusion fronts. Descending fronts propagate with the greatest vclocitics, its much as 80 times more rapidly than the reaction-diffusion front. Using the theory of multicomponent (double-diffusive) convection, we can explain this behavior in which there is a thermally induced density decrease in the I'ront as well as an isothermal density increase. We qualitatively account for the observed concentration and orientation dependences. ah
Introduction In ii previous work,' we analyzed the stability of density gradients induced by traveling fronts, as well as the mechanisms of free convection, which may greatly affect the front velocity. Density gradients are caused by the heat released or absorbed, a thermal effect. The difference in the partial molal volumes of thc reactants and products can also result in an isothermal density change. We determined that there are two different mechanisms by which convection may increase the front velocity, depending on the relative signs of these two effects: simple convection and multicomponent (double-difrusive) convection. I n thc prcccding work, we have considered simple convection in the iodate-arsenous acid system.2 In that system, simple convection increased the propagation velocity for both ascending and horizontal fronts. Descending fronts were unaffected. This type of convection is the cause of smoke rising and the increase in the rate of a burning cigarette held filter side u p 3 Bazsa and Epstein4 observed in the iron(1l)-nitric acid system that in a vertical tube descending fronts propagate more rapidly than ascending ones. What is most surprising is that not only do the descending fronts propagate more rapidly than the ascending ones, and necessarily the pure reaction-diffusion fronts, the ascending fronts also propagate more rapidly than the pure reaction-diffusion fronts. Simple convection is clearly not the operative mechanism. A mode of convection that increases front velocity for both ascending and descending fronts is required, even when the density gradient is statically stable. A type of convection that can occur even with an apparently stable density gradient is called Currcnf nddrcs5. Department of Chemistry, University of Southern Mississippi. tiattioburg, MS 39406-5043. Kossuth Lnjos University.
'
0022-365419 112095- I 306$02.50/0
multicomponent or double-diffusive convection. In this work, we present the results of experiments on the iron(1l)-nitric acid system and, using the theory of multicomponent convection, provide qualitative explanations for the dependence of the velocity on initial concentrations, tube size, and tube orientation. Chemistry of the Iron(l1)-Nitric Acid System The oxidation of Fe(l1) in nitric acid proceeds according to the following r e a c t i o d
2Fe2+
+ 3H+ + NO1- = 2Fe3+ + H N O z + H 2 0
(I)
An intermediate is FeN02+, which is brown and forms the basis of the brown ring test for the qualitative analysis of nitrates and nitrites6 The kinetics of this reaction have been studied in batch' and in a flow reactor' and found to be autocatalytic in nitrous acid. Bazsa and Epstein4 determined that this system can support traveling waves in which a brown ring of FeNOZf propagates in a tube. In convection-free systems, the ring is extremely narrow (=0.1 cm). P6ta and Lengyel* obtained an approximate analytical expression for the convection-free front velocity as a function of ( I ) PoJman, J. A.; Epstein, I . R. J. Phys. Chem. 1990, 94, 4966. (2) Pojman, J. A.; Epstein, I. R.; McManus, T. J.; Showalter, K. J . Phys. Chem., preceding paper in this issue. (3) Pojman, J. A. J. Chem. Educ. 1990, 67, 792. (4) Bazsa, G.; Epstein, I . R. J . Phys. Chem. 1985, 89, 3050. ( 5 ) Orbln, M.; Epstein, 1. R. J. Am. Chem. SOC.1982, 104, 5918. (6) Swift. E. H . A System of Chemical Analysis for the Common Elements; W. H . Freeman: San Francisco, 1939; pp 485-486. (7) Epstein, I. R.; Kustin, K.; Warshaw, L . J . Am. Chem. SOC.1980, 102, 375.
(8) Pdta, G.; Lengyel, I.; Bazsa. G . Submitted for publication in J. Phys. Chem.
0 1991 American Chemical Society
Convective Effects on Chemical Waves. 3
The Journal of Physical Chemistry, Vol. 95, No. 3, 1991
the reactant concentrations. Pojman et al.9 have recently performed simulations using an eight-variable mechanism5 to reproduce reasonably well the convection-free experiments.
Experimental Section Calorinzetry. Calorimetric measurements were carried out using a simple Dewar flask with a Beckman thermometer. To determine an accurate enthalpy, the specific heat of 3 M nitric acid was measured and found to be 3.72 f 0.1 1 J/(g "C). This is in good agreement with an estimate of 3.71 based on a value of 1.74 J/(g "C) for pure HNOI.'" Measurements were performed on 200 cni' of rcaction mixturc of 3.0 M nitric acid and 0.20 M Fe(NH,)2(S0,)I.hH,0 into which a drop of concentrated sodium nitritc solution was added. A 2.1-2.2 OC increase in temperature of the mixture was observed within a few seconds. The enthalpy of the rcnction was determined to be -1 16 f 9 kJ/mol. Isothermal Density Change. Density measurements were made on unreacted and reacted solutions, using a pycnometer at 25 f I "C. The reaction was allowed to proceed and then cool to ambient temperature. With [Fe(ll)],, = 0.2 M, three measurements were made with different nitric acid concentrations:
[ HNO,], = 1 .O M; Apt, = 0.00047 g/cm3 [HNO,], = 3.0 M; Ap, = 0.00060 g/cm3 [HNO3]n = 4.0 M; Apu = 0.00055 g/cm3 Wc used thc avcragc of these three values, Ap(. = (5.4 f 0.6) X 10 g/cm3. Proparotion of Solutions. All solutions were prepared in glassware cleaned with chromic acid. Commercial 65% nitric acid was diluted to 8 M and then purged intensively in a brown bottle for I .5 h with argon. Trace amounts of H N O ] were scavenged by :idding enough N H 2 N H ? . H 2 S 0 4to make a solution with a concentration of M. I f a brown color persisted after purging with argon. then 0.1 g of NaN, was added to 1000 mL of the nitric acid solution and the purging repeated. The initiating solution consisted of 0.05 M N a N O z in 3 M HNO,. Thc nitric acid was not purged. Thc Fc(ll) solution was prepared in a 50.0-mL volumetric flask by dissolving 9.804 g (0.0314 mol) of Fe(NH4)?(S0,)?.6Hz0 in a mixture of S.0 mL of I .O M H,SO, and 35.0 mL of distilled watcr and thcn dilutcd to a volume of 50.0 mL. A 10.0-mL solution W;IS prcparcd immcdiately before measurements were performed by mixing 0.50 mL of 2 m M N H 2 N H y H 2 S 0 4solution and sufficient amounts of the Fe(ll) and nitric acid solutions. Mea.suroircwt.yof Wove Front Velocities. The measurements were performed in a clean buret tube (of Pyrex glass) by drawing up the Fe(ll)-nitric acid solution with a pipet bulb. Waves were initiated by adding a drop of the initiator solution to either the bottom or the top of the tube. The time required for the brown ring to propagate I .0 cm was recorded. All measurements were repeated six times and the values averaged.
Results Observations of the fronts indicate several qualitative features: A.vcending Fronts. ( I ) Convective fluid flow is visible above thc front, which appears as a brown ring with a fuzzy brown region above the front: the width is proportional to the velocity. (2) The velocity is greater than that of the pure reaction-diffusion front. (3) The velocity is greater in wide tubes; a critical tube radius exists below which the front propagates at the same velocity as the react ion-di ffusion front. Dm-ending Fronts. ( I ) Convective fluid flow is visible above the front. ( 2 ) The front is very wide and diffuse: the width is proportional to the velocity. For the fastest fronts, the width can be a s large as 2 cm. (3) The velocity is as much as 10 times greater than that of an ascending front. The velocity is greater in wide ( 9 ) Pojman. J. A.; Epstein, I . R.; Karni, Y . ; Bar-Ziv, E. J. Phys. Chew., in press. ( I 0) D'Ans-Lax Tuschenbuch f u r Chentiker und Physiker; SpringerVerlag: 1967: Band I .
i
40 -
0
Descending (w) Descending (a)
30
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0
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Ascending (w) Ascending (a)
10
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0 0 04
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0.05
0 06
0 07
0 08
Radius (cm) Figure 1. Vclocity as a function of tube radius for tubes surrounded by water ( w ) and air (a). Error bars for the ascending fronts are smaller than the circles, which also overlap. [Fe(ll)], = 0.20 M, [HNOJ, = 3.0 M. For r = 0.046 cm, wall thickness (WT) = 0.238 cm; r = 0.050 cm, WT = 0.245 cm; r = 0.070 cm, WT = 0.222 cm. The pure reaction-diffusion front4 has a velocity of 1.2 X IO-' cm/s.
tubes, and there is a critical radius below which the front is a pure reaction-diffusion one. Horizontal Fronts. No detailed observations were made. Effect of Medium Surrounding Tube. Nagyp5l et al." observed that wave front velocity in the chlorite-thiosulfate system was affected by changing the medium surrounding the tube from air to water. We performed all experiments in both configurations. Within the experimental uncertainty, there was no difference in any experiment. The results of one experiment are shown in Figure I . We cannot conclude, however, that the solution is adiabatic. Given the small temperature differences involved and the relatively large thickness of glass, it is most likely that the glass acts as a thermal buffer between the solution and the surroundings. I f the reaction were run in thinner walled tubes, especially of a material with a low heat capacity, the front velocity would probably be affected by the surrounding medium. Henceforth, we will only show data from the tube surrounded by air. Effeect of Tube Radius. Figure I shows the front velocity as a function of tube radius. For the same concentrations, Bazsa and Epsteid determined that there is a critical radius at which the ascending and descending fronts have the same velocity. The critical radius is 20.03 cm for the type of glass employed. They also measured the convection-free velocity, which is 1.2 X cm/s. In the widest tube, it can be seen that the descending front has a velocity 25 times as great as the convection-free front. Effeect of Tube Orientation. The angle of the tube with respect to gravity was varied. We define Oo as a descending front in a vertical tube. An ascending front in the same tube corresponds to 180". Figure 2a shows the velocities of fronts in the 90-1 80" range, Le., horizontally propagating fronts to ascending fronts. Notice that at 135O there is a maximum. The descending fronts do not exhibit a maximum velocity between 0' and 90' as can be seen in Figure 3. In fact, the velocities of descending fronts are proportional to the square of the cosine of the angle. Effects of Initial Concentrations. In Figure 3, we see that both the ascending and the descending front velocities increase monotonically with the initial concentration of Fe(1l). Experiments with higher concentrations were not reproducible because the width of the fronts became so large, and the amount of convection so great, that accurate measurements were not possible. Figure 4a shows the ascending front velocity as a function of nitric acid concentration. The velocity increases monotonically with increasing nitric acid concentration. Measurements at higher concentrations were not possible because the solutions were unstable. Figure 4b demonstrates that there is a maximum in the descending velocity a t 2.5 M nitric acid. ( I 1 ) Nagypll, 1.; Bazsa, G.; Epstein, 1. R. J . Am. Chem. SOC.1986, 108, 3635.
Pojman et al.
1308 The Journal of Physical Chemistry. Vol. 95, No. 3, 1991 d
. 6 h
In
v
0 0 0 7
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-0
8 120
100
80
160
140
180
200
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[Nitric Acid] b
40 T
. 6
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v)
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0 0 0 7
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.-0
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8
0
0.2
0.0
0.6
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Cos2 (Angle) Figure 2. Velocity as a function of tube orientation: (a) ascending, (b) dcsccnding. fFe(ll)],, = 0.20 M. [HN0310= 3.0 M. r = 0.07 cm, W T
= 0.221 CI11.
- -
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I
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'
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[Fe(ll)l Figure 3. Velocity of ascending and descending fronts in a vertical tube as ;I function of [Fe(ll)],. [HN0310= 3.0 M. r = 0.07 cm, W T = 0.222
cm. Discussion Prwim.v/v Propmen Mechanisms. Given that the reaction is exothermic. one might expect that thermally induced convection would cause the same behavior observed in the iodate-arsenous acid rcaction.'.'2 where the ascending velocity is faster than the descending one. Bazsa and Epstein4 demonstrated that convection affects thc wavc vclocity whcn they eliminated the anisotropy betwccn asccnding and descending fronts by performing experi-
1
2
3
4
5
6
[Nitric Acid] Figure 4. Velocity as a function of [HN0310: (a) front ascending in a vertical tube, (b) front descending in a vertical tube. [Fe(ll)lo = 0.20 M. r = 0.07 cm, WT = 0.222 cm.
ments in a narrow tube (r = 0.03 cm) and in tubes filled with silica gel. They incorrectly attributed the slower ascending fronts to convection-induced mixing which interfered with nucleation of the autocatalytic reaction. Such a mechanism cannot explain how the convection-free fronts proceed most slowly. Nagypil et al.lI correctly identified the key difference in this system which distinguishes it from the ones affected by simple convection. They measured the isothermal density difference caused by the change in chemical composition. They found that g/cm3 when a 0.2 M there was a density change of 1.7 X Fe(l1) solution was oxidized by 3 M HNO,. Such a change cccurs because the partial molal volumes of the products are smaller than those of the reactants.I3 Nagypil et al. also studied the velocity dependence on the orientation of the tube in which the chlorite-thiosulfate wave propagates. They observed that, under certain conditions, the ascending front was faster than the descending and pure reaction-diffusion fronts. However, by reducing reactant concentrations, it was possible to reverse this behavior so that the descending fronts were faster. They were also able to achieve this inversion by immersing the tube in water instead of air. The chlorite-thiosulfate reaction is extremely exothermic (AH' = -1077 kJ/mol).I4 The presence of water, with its greater thermal conductivity, served to remove enough heat to allow another mechanism of convection to operate. Nagypil et al. also studied the velocity dependence on the angle of the tube with respect to gravity. The ascending front shows a maximum at about 135' under conditions in which the ascending fronts were fastest. ( I 3) For a more detailed discussion on factors affecting the density of a solution, see ref I as well as the preceding work in this issue.
(12) Hanna. A.; Saul. A,; Showalter, K . J . Am. Chew. Soc. 1982. 104, 3838.
( 1 4) Barner. H. E. Handbook of Thermochemical Data for Compounds and Aqueous Species; Wiley: New York, 1978.
Convective Effects on Chemical Waves. 3
The Journal of Physical Chemistry, Vol. 95, No. 3, 1991
I
HOT, SALTY WATER
1
FORCE
1309
OLD, FRESH WATER
I
COLD, FRESH WATER
Figure 5. Mechanism which causes double-diffusive convection in the finger regime. Apc and APT have opposite signs, and the net density gradient appears to be stable. However, if a small parcel of the warm, salty solution enters the lower section, because the heat diffuses faster than the salt, the parcel is left with a greater density than the surrounding iiicdium. Tlic buoyunt forcc pulls it down.
Stubilitj-h d j x i s . Wc mcasured an isothermal density change (A/)()of (5.4 f 0.6) X g/cm3, with [Fe(l1)lo = 0.2 M, and a thcrmnlly induced dcnsity change (Apr) of -5.6 X lo4 g/cm3. Wc usc Turncr’s stability ratio,” which we have employed previously
R,, = -Apr/Ap,. = -AHo/@C,, x 1
(2)
whcrc (j is thc mean molar isothcrmal density coefficient.’ Noticc that thc stability ratio is independent of concentration for an adiabatic system but only depends on the two thermodynamic quantities. the reaction enthalpy and the reaction volume. However, if heat is lost to the walls and the environment, then Ap7 will depend on the rate of reaction. For the iron(I1)-nitric acid reaction, R,, is about equal to unity, so the net density change is approximately zero. Thus, as the front proceeds, it causes no significant density change in the solution. To understand how convection can occur, we must consider an analogous configuration with both salt and thermal gradients. We replace the product and reactant gradients by a single gradient of salt having the same density gradient as that induced isothermally by the traveling front. To simplify matters further, replace the smooth gradient occurring in the front by a sharp interface between solution layers. This approximation does not change the fundamental considerations. Now consider hot, salty water above cold, fresh water as depicted in Figure 5. The system appears to be stable, if R,, = l (no net density gradient). Yet, it may not be. Imagine that at the interface between the layers a small parcel of the upper solution were to deviate from its position by gradually enlarging and descending into the cold, fresh region. Because the temperature and concentration are higher than in the surrounding region, heat and salt will diffuse out. The heat will leave at a greater rate, because of the larger diffusivity of heat (about 2 orders of magnitude). Now the parcel is cool and dense; because of its higher concentration, it sinks. Further, the surface area of the parcel increases, accelerating the loss of heat and the corresponding increase in density. Similarly, if a parcel of cold, fresh water protrudes into the hot, salty layer, heat will diffuse in faster than the salt. This will leave the parcel less dense than the surrounding layer, and the parcel will rise. What results are known as “salt fingers”, which appear as long slender regions of alternately desccnding a n d asccnding fluid. I f cold, fresh water ovcrlics dcnscr, hot, salty water, the system is still statically stablc. Howcvcr, wc again must consider a perturbation at the interface, as depicted in Figure 6 . The heat diffuscs faster than the salt, leaving the parcel heavier than the surrounding region. It sinks and can continue through the interface into the hot, salty region, where it again heats up and rises. This oscillatory behavior will continue as long as the density gradients persist. This oscillatory process increases the effective area of the intcrfacc, thcrcby increasing the flux of salt. The heat released into the region above the interface causes convection, which further increases the mass transport. Both ol‘ thc above types of convection are categorized as “doublc-diffusivc” convcction (“thermohaline” when the two
Figure 6. Mcchanism which causes double-diffusive convection in the diffusive regime. Apc and APT have opposite signs, and the net density gradient appears to be stable. The parcel of hot, salty solution which enters the cold, fresh region above loses heat, becoming more density than the surrounding region. It sinks back to the hot region and regains heat.
components are salt and heat) and, more generally, “multicomponent c o n ~ e c t i o n ” . ~ It ~ .is~ ~not necessary that the two components be a solute and heat. Any two materials with different diffusivities can cause these phenomena, even if the differences are small as with salt and sugar or with polymer solutions of different molecular weight distributions.” This phenomenon has been extensively studied by oceanographers because of the role it plays in ocean current mixing.’s*’6.’X-”It has also been shown to cause pattern formation in some photochemical reactions.22 The two behaviors we have described correspond to the “fingering” and “diffusive” regimes of double-diffusive convection, respectively. The stability of these two arrangements has not been thoroughly studied for solutions in a narrow cylinder, although some preliminary work has been done.I3 Turner has shown that there is a critical value of R, = 1, as in the iron(I1)-nitric acid case. From our work on simple convection,2 we know that there is a critical tube radius for the onset of convection. A critical radius has also been observed in the iron(I1)-nitric acid ~ y s t e m . ~ Ascending Fronts. The ascending front produces a configuration corresponding to the “diffusive regime”, with hot, salty water underlying cold, fresh water because the reacted solution is warm and, isothermally, more dense than the unreacted solution. Experiments with a stable salt gradient heated from below indicate that fluid flow occurs above a spontaneously formed interface, caused by the heat released by the oscillating parcels of fluid. The convection increases the mass transport as does the increased surface area caused by the oscillations. The effective mass transport coefficient of salt may be 10 times as great as the diffusion coefficient.I5 Because the wave velocity is proportional to the square root of the autocatalyst’s ( H N 0 2 ) transport coefficient, the ascending front’s velocity should be about 3 times as great as that of the pure reaction-diffusion f r ~ n t . ~ . ’ ~The . * ~data in Figure 1 agree with this order of magnitude prediction. The flux measurements were made in very wide vessels to effectively eliminate any effects of container size, but the actual front velocity is also a function of the tube radius (Figure 1). The observations of the front itself are in accord with the nature of a diffusive interface. Turnerts observed that visible fluid flow occurs above the interface, which is the cause of the “fuzzy” appearance of the front. The width of the brown ring increases with increasing velocity because the increased convection enlarges the front. ( 1 5) Turner, J. S . Buoyancy Effects in Fluids; Cambridge University Press: Cambridge, 1973; Chapter 8. ( 1 6 ) Turner, J. S . Annu. Reo. Fluid Mech. 1985, 17, 1 1. (17) Nason, P.; Schumaker, V.; Halsall. B.; Schwedes, J. Biopolymers 1969, 7, 241. ( 1 8) Huppert, H. E.; Turner, J. S . J . Fluid Mech. 1981, 106, 299. (19) Huppert, H. E.; Manins, P. C. Deep-sea Res. 1973, 20, 315. (20) Linden, P. F. Deep-sea Res. 1973, 20, 325. (21) Turner, J. S . Deep-sea Res. 1967, 14, 599. (22) Avnir, D.; Kagan, M. Nature 1984, 307, 717. (23) Georgopoulos, E.; Horwitz, J. A.; Rosenblat, S. Acta Mech. 1987, 70, 177. (24) Tyson, J. In Oscillations and Traveling Waves in Chemical Systems; Field, R. J., Burger, M., Eds.; Wiley: New York, 1985.
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T h e Journal of Physical Chemistry, Vol. 95, No. 3, 1991
Thc ascending velocity exhibits a maximum at 1 3 5 O and then dccreases as we move away from the vertical to a value exceeding the value at 180’ (Figure 2a). The maximum probably occurs at 135’ because the surface area of the front is greatest at that oricntation. The greater the surface area, the greater will be the moss transport as the fluid moving by convection sweeps along the front. The velocity is greater in the horizontal orientation ( 9 0 O ) than in the vertical ( 1 80’) because a horizontal density gradient is niorc tinstnblc than a vertical one.’.2 The iiionotonic increase in the velocity with increasing [ Fe( II)], is interesting because in convection-free experiments a monotonic dccrcnsc has been observed.x The velocity increase must be due to convection. Increasing [Fc(ll)],, increases l p , which is the driving rorcc for the convection above the front in the diffusive regime. A monotonic increase in velocity is also observed with increasing nitric acid concentration (Figure 4a). Convection-free fronts exhibit similar behavior in experimentsBand simulation^,^ but with much smaller velocities. The values of Ap< and Ap, are not affected by the nitric acid concentration. Why does the velocity incrcnse? At higher nitric acid concentrations, the reaction occurs morc riipidly. The Limount of heat liberated is the same as at lower concentrations. but the thermal gradient is larger because the heat has less time to diffuse. Also, the faster the heat is released, the sm:illcr thc pcrccntagc of hcat absorbed by the wall. The maximum tciiipcraturc and thc thcrmal gradient are larger. The larger the theriiial gradient, the morc convection occurs above the front and the greater the mass transport.25.2h This increased convection-misted 1ii;iss transport is thc origin of the increased front velocity . Dc.vcwitliug Frotit. The descending front produces a configuration corresponding to the “finger regime”, with hot, salty water ovcr cold. fresh water. because the reacted solution is warm and, isothermally. morc dcnsc than the unreacted solution. In expcriments on heated salt water, in a solution with a statically stable density gradient, long alternating “fingers” of fresh and salty water arc observed to move rapidly across the interface.ls~’h~’8-2’ From expcrimcnts on heated sugar solutions, we can estimate the increase in mass transport from finger convection. Linden2” has measured the solute f l u x in such solutions and calculated an effective mass transport cocfficicnt that is 4 ordcrs of magnitude greater than the diffusion cocfficicnt! Turner has estimated that the effective transport coefficient for salt in a solution in which ApC= 4 X (g/cm’) and R,, = 1 . I is 5 orders of magnitude greater than the diffusion cocfficicnt of salt.?’ This increase in mass transport rcsults both from thc incrcasc in surface area provided by the fingers as compared with a flat interface and from the macroscopic movement of fluid in the fingers themselves. A 4 ordcr of niagnitudc incrcasc in the transport coefficient of the autocatalyst would increase the front velocity by a factor of 100. The measurements of Turner and Linden were performed in large containers. so the results provide a limiting value. We obscrvcd increases in velocity of 30 times over the reaction-diffusion case. I n even larger tubes, Bazsa and Epstein4 observed increases of almost I00 times. Thc qualitntivc obscrvations of thc fronts are in accord with a fingering convection process. The very wide fronts result from thc rapidly moving fingers of reacted solution. Alternating, ascending fingers of unreacted solution are not visible because the unrcactcd solution is clear. I f an inert dye were added to the solution. thcn such fingers should be visible. The vigorous fluid motion above the front is a result of the macroscopic motion of thc fingers. For large velocities, the experimental uncertainty increascs (Figures I , 3. and 4) as the front becomes more difficult to locate visually and because the convective fluid becomes turbulent. Thc dcpcndcncc of the descending velocity on oricntation in Figure 2b shows a lincar dcpcndcncc on the square of the cosine of the angle. We cxpcct that the vclocity of the fingers should (25) Turner, J . S. I n t . J . Heat Mass Transfer 1965. 8. 159 (26) Turncr. .I.S. J . Fluid Mech. 1968, 33. 183.
””
1
0
0
Thermal
1
2
3
4
5
6
[Nitric Acid] Figure 7. Thermal and concentration gradients as a function of [HNO& determined by simulations (scc ref 9 for dctails). Gradicnts wcrc determined manually from plots of tcmpcraturc or [Fc(ll)] vcrsus position and have units OC/cm and M/cm, respectively.
depend on Ap, and on the magnitude of the component of the gravitational vector along the axis of the tube, a function of the cosine of the angle: F = g cos 8 Why the velocity should be a function of the square of the cosine is not clear. The monotonic increase in velocity with increasing initial Fe(ll) concentration is consistent with a fingering mechanism. As [Fe(ll)], increases, so does Ap
