Acceleration of chemical reaction by chaotic mixing

4 Pages·0.12 MB·English

by  M. Chertkov

Acceleration of chemical reaction by chaotic mixing
M. Chertkova and V. Lebedeva,b
a Theoretical Division, LANL, Los Alamos, NM 87545, USA
b Landau Institute for Theoretical Physics, Moscow, Kosygina 2, 117334, Russia
(Dated: January 10, 2003)
Theory of fast binary chemical reaction, + , in a statistically stationary chaotic flow
A B → C
at large Schmidt number Sc and large Damk¨ohler number Da is developed. For stoichiometric
conditionweidentifysubsequentstagesofthechemicalreaction. Thefirststagecorrespondstothe
exponentialdecay, exp( λt)(where λis theLyapunovexponentof theflow),of thechemicals in
∝ −
the bulk part of the flow. The second and the third stages are related to the chemicals remaining
in the boundary region. During the second stage the amounts of and decay 1/√t, whereas
3 A B ∝
thedecay law duringthe third stage is exponential, exp( γt),where γ λ/√Sc.
0 ∝ − ∼
0
PACSnumbers: 47.70.Fw,47.27.Qb
2

n
a A common expectation is that random advection the flow, which is assumedto be incompressible ( v =
J ∇·
should essentially accelerate chemical reactions in fluid 0), κa,b are the diffusion coefficients of the chemicals.
7
phase, since it should lead to homogenization of the re- Here, we assume that the fluid dynamics is independent
2
action mixtures. Then dynamics is determined by an ofthechemicalreaction,thatisthevelocityfielddoesnot

] interplayofthreefactors: diffusion,advectionandchem- sense changes in the chemical concentrations nor heat
D
icalreactionrate. Typicalsituationrealizinginchemical is released in the result of the chemical reaction. Our
C reactors is that chemical reaction itself is much faster approachisalsoapplicabletothecase,realizedintubular
. than mixing and diffusion, i.e. the Damk¨ohler number, chemical reactors, when the solution of the chemicals,
n
Da, which is defined as the ratio of the mixing time to preparedatthe entrance,is then pushedthrougha pipe.
i
l
n the characteristic time of the reaction [1], is large. For In this case the position along the pipe plays the role of
[ the binary reactionthis separationoftemporalscalesre- time in the decay problem.

sults in formation of lamellar structure, build of stripes, The major question addressed in the letter is: how do
1
v populated solely by one chemical. The stripes of differ- the total amounts of chemicals, Na,b = R drna,b(t,r),
7 ent chemicals are separated from each other by an in- decay as time t advances? We focus primarily on the
3 terface of complicated shape, and the chemicals co-exist stoichiometric case N = N . This case is of major in-
a b
0 only in the narrow interface domain where the chemical terest for applications, as it allows to get pure product
1
reaction occurs. The reaction is limited by diffusion in (not mixed with the reagents), by the time reaction
0 C
the sense that diffusion controls fluxes of the chemicals is completed. (An effect of a mismatch between N and
3 a
0 into the interfacial reaction zone [2, 3, 4]. The physical Nb is briefly discussed later in this text.) We identify
/ picture of the acceleration due to the random advection majorstagesof the chemicalreactionandrelate them to
n
is that it stretches domains populated by one chemical the chemicals decay in different parts of the flow. An
i
l
n into thin sheets, so that the chemical reaction driven by essentialpartof the evolutionis relatedto the boundary
: diffusionproceedsmoreefficientlybecauseofanessential region.
v
increase of the interface area. In this letter we explain We discuss mainly the case of κ = κ = κ. (It is
i a b
X howthisgeneralphysicalpicture,formulatedinitiallyfor argued later in the text that κ = κ does not lead to
a b
6
r unbounded flows applies to chaotic flows confined to a significant changes in the theory.) Then one obtains a
a
finite geometry. closed equation for the difference field, n=n n ,
a b
−
We consider a binary chemical reaction, + ,
in a dilute solution of two chemicals. We AstudyBd→ecaCy ∂tn+(v )n=κ 2n, (2)
·∇ ∇
problem, with an initial distribution of the chemicals
A from Eq. (1), i.e. one finds that n(t,r) is a passive
and , created by injecting solution of one chemical, say
B scalar field. Note, that n has no definite sign, and that
of ,intosolutionoftheotherchemical, . Itisassumed
thaAt the inverse reaction + iBs negligible, i.e. R drn=0 in the case of perfect matching, Na =Nb.
C → A B Weassumethatthechaoticstatisticallysteadyvelocity
there is no back influence of on the distribution of
C A field v(t,r) contains only few harmonics of the reservoir
and . Then molecular concentrations of the chemicals,
B size, i.e. the flow is smooth. This regime can be realized
n and n , vary according to the following non-linear
a b in chemical reactors with mechanically rotating mixers
governing equations (see, e.g., [5])
orexternallydrivenmagnetsstirringthefluidintheper-
∂ n +(v )n =κ 2n Rn n , (1) fect mixing devices and also in the tubular reactors at
t a,b a,b a,b a,b a b
·∇ ∇ − moderate Reynolds numbers. (See [6] for a discussion of
whereRisthereactionratecoefficient,visthevelocityof the chemical engineering principles behind various reac-
2
tordesigns.) Complementarytoitspracticalsignificance,
A B
passive scalar advection in a smooth chaotic flow is also B
A A
a well studied (by both theoretical [7, 8, 9, 10, 11] and
B
experimental [12, 13, 14] means) subfield of statistical
peripheral
hydrodynamics. (See also reviews [15, 16].) The pas-
sive scalar decay theory, developed in [11, 17] for an un-
bounded flow, was recently modified for bounded flows, bulk
i.e. for chaotic flows with suitable (no slip) conditions r
d
n n
on the boundary [18]. Smoothness of the flow allows b a
one to approximate the velocity difference between close
points by a linear, although fluctuating in time, profile. boundary
In the bulk region, the linear profile approximation is L rbl rch
valid for separations smaller than the system size L. In
the periphery,i.e. closetothe solidboundary (wall),the
linear profile approximation is valid for velocity fluctua- FIG. 1: Chemical reactor boundaryis drawn by solid line on
tions on a scale smaller than distance to the boundary. the main chart. Dotted lines separate bulk, peripheral and
boundarydomainsoftheflow. Chartintheupperrightcorner
An important consequence of the linear velocity profile
shows schematically the stripe structure magnified from the
approximation is that close Lagrangian trajectories di-
bulkand/orperipheraldomains. Blackandwhiteregionsare
verge exponentially in time. The mean logarithmic rate
theonespopulatedby and respectively. Thechartinthe
of the nearby Lagrangian trajectories divergence defines low right corner showsA(underBeven stronger magnified glass)
theLyapunovexponentoftheflow,λ. Notice,thatinthe distribution of chemicals normal to their contact interface.
peripheral domain advection is essentially anisotropic,
and the stretching rate along the boundary is estimated
byλ, while the stretchingrateinthe directionnormalto colored. To resolve the interface domain, even stronger
the boundary is significantly smaller. magnification is needed. Dependence of the chemicals
We now discuss characteristic spatial scales in the concentrationson the coordinate normalto the interface
problem. The size of the system, L, which is also the is shown schematically on the chart in the lower right
chaotic flow typicaleddy scale, is the largestscale in the corner of the figure.
problem. A comparison of the advection and diffusion The separation of scales, rch rd,L, allows an im-
≪
terms in Eq. (1) sets the dissipative scale of the flow, portant simplification in the description. Indeed, the
which in the bulk region is rd = pκ/λ. We assume chemical reactiontakes place in the rch-narrowinterface
that the Schmidt number, Sc (L/rd)2, is large, i.e. in domain, where the values of na and nb are comparable.
∼
the asymptotically wide range of scales, L r r , Outside this narrow region, i.e. in the region dominated
d
≫ ≫
advection dominates diffusion. The width of the diffu- by one of the chemicals, the presence of the other chem-
sive boundary layer is estimated by rbl ∼ Sc1/4rd, i.e. ical is negligible. Thus, in the limit rch → 0, i.e. when
r > r . Yet another important scale, associated with the reaction zone becomes infinitesimally thin, one ob-
bl d
the chemical reaction itself, is the size of the reaction tains na = n, nb = 0 for n > 0 and na = 0, nb = n
−
zone, r (the width of the interfacial domain where the for n < 0. These relations imply a remarkable conclu-
ch
chemical reaction occurs). In the bulk region the scale sion[2,4]: the fastchemicalreactioncanbe describedin
is estimated by r = r [λ/(Rn )]1/3, where n is a termsofthelinearsetting(2)whichdoesnotcontainthe
ch d m m
typical concentration of the chemicals inside the layers. chemical reaction rate coefficient R. The reaction rate
(The estimate for the width of the reaction zone should is determined by the diffusion fluxes of and to the
A B
be modified near the boundary, where it appears to be n = 0 surface. These fluxes are equal to each other and
larger than in the bulk.) Initially, rch is much smaller opposite in sign, which is translated, at rch → 0, into a
than r (and, consequently, than L); the inequality is a continuity condition for n at the interface. This obser-
d ∇
consequence of the Da 1 assumption. (Indeed, in ac- vation also means that, while rch is much smaller than
≫
cordancewiththe definition,the Damk¨ohlernumbercan all other relevant scales, our problem is reduced to the
be estimatedasDa Rn0/λ,wheren0 is atypicalvalue problem of scalar decay in chaotic bounded flow.
∼
of the initial chemical concentration. Thus, at t = 0, Furthermore,fromEq. (2),itis straightforwardtode-
r Da−1/3r .) However, r grows as n decreases. rive equations for correlation functions of n (the deriva-
ch d ch m
∼
Thus even though the separation of scales is perfect ini- tionprocedureissimilartotheonedescribedin[10,11]).
tially,iteventuallybreaksdownatthelateststageofthe Theequationforthemeanvalueofn, n ,derivedbyav-
h i
chemical reaction. A cartoon illustration of the scale hi- eragingovertimeslargerthanthecorrelationtimeofthe
erarchy is shown in Fig. 1. The magnified striped struc- flow τ0, is
ture is shown on the chart in the upper right corner of
2
thefigure. Regionspopulatedbyonechemicalaresingle- ∂ n = (D n )+κ n . (3)
t α αβ β
h i ∇ ∇ h i ∇ h i
3
In the case of a short-correlated(in time) flow, λτ0 1, in the peripheral domain, where advection is more effi-
≪
the turbulent diffusion tensor D is expressed through cientinthedirectionalongtheboundarythaninthenor-
the velocity pair correlation function: D (r) = mal direction. This anisotropy causes the layers in the
αβ
R0∞dthvα(t,r)vβ(0,r)i. Eq. (3) remains valid in the peripheraldomaintostretchmainlyalongthe boundary.
general, not necessarily short-correlated limit. However, Thestripesclosertotheboundaryshrinkslowerthanthe
the relation between the eddy-diffusivity tensor and the remote ones, since the normal to the boundary compo-
velocity correlations becomes more complicated. D(r) nent of the stretching rate decreases as one approaches
tends tozerowhenr approachesthe boundary,sincethe the boundary. Thereforethe developedlayeredstructure
flowvelocitytends to zerothere. (The longitudinalcom- (i.e. the one which contains stripes of the diffusive scale
2
ponent D of the tensor D behaves as D q , whereas width)occupiesapartoftheperipheralregionwherethe
k k
itstransversecomponentD behavesasD∝ q4,where amountsof and becomenegligible. Thus,theempty
⊥ ⊥
∝ A B
q is the distance to the boundary.) Our description of ofchemicalsregion,formedinthe bulk by the end ofthe
the chemical reaction problem is based on the solutions first stage, starts to expand towards the boundary. As
of Eq. (3) in different spatio-temporal domains. Know- a result, the chemicals are arranged mainly within a δ-
ing n , one can establishthe temporalbehavior ofN . vicinity of the boundary (wall), δ L/√λt, where the
a,b
h i ∼
The brief style of this letter does not allow us to present concentrations of the chemicals remain practically un-
the complete analysis here. Therefore, below we report changed. Outside this layer, at L q δ (where q
≫ ≫
finalresults,omittingdetailsofthederivation. Toclarify is the separation from the boundary), the concentration
the results, we also pay special attention to presenting a of chemicals decreases algebraically n t−3/2q−3.
a,b
h i ∝
physical picture of the phenomenon. During this stage the overall amounts of chemicals de-
We find that the chemical reaction (which starts at crease as δ(t), that is 1/√t. The spatio-temporal pic-
∝
t=0) undergoes the following four stages: ture explained above follows from the universal form of
I. Formation of stripes in the bulk. Advection cre- thevelocityfieldprofileintheproximityoftheboundary.
atesfromaninitiallysmoothdistributionastripedstruc- This stage lasts for τ2 √Sc/λ, i.e. until δ shrinks to
∼
ture of alternating domains of and [4]. The stripes the width of the boundary layer, rbl.
A B
become dynamically thinner, i.e. inhomogeneities of III. Boundary-layer-dominated dynamics. Chemicals
smallerandsmallerscalesareproduced. Once the width remainmainlywithinther -thin(notvaryingwithtime)
bl
of the stripe decreases down to the diffusive scale, r , vicinity of the boundary. The boundary layer width, r ,
d bl
the stripe collapses (wiped out by the diffusion-limited isstillmuchlargerthanthereactionzonesize(definedfor
chemical reaction) in a time λ−1. Since the stretching the boundary region), so that the passive scalar descrip-
∼
(contraction)process leading to creationof the stripes is tion applies. The interfacial area where the chemicals
exponential in time [7, 8, 9, 10, 15, 16], the initial stage interact does not change significantly anymore. Thus,
(when the rd-stripes are formed) lasts for τ1 ln(Sc)/λ, due to linear relation between flux of chemicals to the
∼
i.e. justthetimerequiredforthecascadeofpassivescalar interface and their concentrations, the algebraic decay
to run from L down scale to r . Even though the inter- switches to an exponential one, i.e. n exp( γt),
d h a,bi ∝ −
facial area increases exponentially during the first stage, for t τ2, where γ λ/√Sc L−1√λκ. (Note, also,
≫ ∼ ∼
N do not vary significantly. By the end of this stage that the slow-exponential regime, derived in [18] for the
a,b
the bulk parts of N begin to decay rapidly (exponen- passivescalar,is consistentwith the experimentalobser-
a,b
tially), with a decrement of the order of λ, according to vations of [14].) Then, N (t) exp( γt). Chemicals
a,b
∝ −
the law ofthe passivescalardecayinanunboundedspa- are mainly concentrated inside the diffusion boundary
tiallysmoothflow[11,17]. Thus,afterthefirststagethe layer. Outside the boundary layer (at q r ) one of
bl
≫
chemicals remain mainly in the peripheral region. the chemicals prevails and its concentrationdecays alge-
Notice also, that after the first stage, stripes of differ- braically, 1/q3. The passive scalar description in the
∝
ent widths, distributed between r and L, are present in vicinity of the boundary layeris brokenwhen r , which
d ch
the bulk. (This multi-scale structure is also seen in the grows exponentially with time, becomes of the order of
passive scalar decay experiment [13, 14], and in binary- r ,i.e. whenattheboundaryn becomes λ/(R√Sc).
bl a,b
∼
reaction numerics [19].) When the r -wide stripe, say, One concludes that the duration of the boundary-layer-
d
of the chemical collapses, then two nearby stripes of dominated stage is τ3 γ−1ln(Rn0√Sc/λ), where n0 is
A ∼
the chemical form one wider stripe. Thus, collapse the initial concentration of the chemicals.
B
of r -narrowstripes is accompanied by creation of wider IV. Nonlinear stage. By the end of the previous
d
stripes, which are shrunk by the flow in turn, and so on stage, advection and diffusion homogenize the remain-
and so forth. ing amounts of the chemicals, first, within the bound-
II. Peripheral-region-dominated dynamics. The same ary layer and later over the entire reservoir. After that
process of layered structure formation takes place in the there are no inhomogeneities of n left in the sys-
a,b
peripheral domain as well. However, advection, which is tem. A purely homogeneous kinetic process takes over:
statistically isotropic in the bulk, is strongly anisotropic dN /dt= RVN N (where V is the chemical reactor
a,b a b
−
4
volume). Thus, N 1/t, during the final stage. isalsorevealed[21]. Afinalremarkconcernsthevalidity
a,b
∝
If N = N then the proposed scheme is valid until of the hydrodynamic description of the chemical reac-
a b
6
N become of the order of N N . Then N satu- tion dynamics. It is known that the character of spatial
a,b a b a
| − |
ratestoaconstant(ifN >N )andN disappearsexpo- fluctuations in the initial distribution of chemicals may
a b b
nentially, exp( RN t/V). (Note that the exponential essentially influence the long-time behavior in diffusion-
a
∝ −
decay starts after a short intermediate stage character- limited chemical systems [22, 23]. In some cases (of low
ized by complete homogenization of due to advection space dimensionality, d 2) large scale renormalization
A ≤
and diffusion.) of the concentration fields due to the small scale fluctu-
Let us now discuss the effect of unequal diffusion co- ations could be important. (See, e.g., [24].) In our case,
efficients, still assuming that pκ /λ L. If Da 1 however,thisdoesnothappenbecausethelong-timecor-
a,b
≪ ≫
then during the first stages, the chemical length r is relations are completely destroyed by chaotic advection.
ch
(as above) much smaller than all other scales. This
problem can also be reduced to a linear one consider- We thankJ.Cardy,G.D.Doolen,A. Fouxon,A.Gro-
ing the advection-diffusion equations in domains popu- isman, I. Kolokolov, V. Steinberg, P. Tabeling, and Z.
latedbydifferentspecies,supplementedbythe condition Toroczkai for helpful discussions and comments.
thatfluxesofthetwochemicalstowardstheinterfaceare
equal. During the first two stages, the evolution is con-
trolled by the stripe formation process which is insensi-
tive to the diffusion. During the latter,thirdandfourth,
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