Ind. Eng. Chem. Res. 2006, 45, 7321-7328
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Onset of Buoyancy-Driven Instability in Gas Diffusion Systems Min C. Kim* Department of Chemical Engineering, Cheju National UniVersity, Cheju 690-756, Korea
Do Y. Yoon Department of Chemical Engineering, Kwangwoon UniVersity, Seoul 139-701, Korea
Chang K. Choi School of Chemical and Biological Engineering, Seoul National UniVersity, Seoul 151-744, Korea
The onset of buoyancy-driven convection caused by gas absorption through the upper free boundary of an initially quiescent, horizontal liquid layer is analyzed under linear theory. It is well-known that convective motion sets in when the Rayleigh number (Ra) exceeds a certain value. In this study, the critical conditions to mark the onset of convective instability are analyzed using the dominant mode method and frozen-time model for large Ra values. The results are compared with those from the propagation theory and also the amplification theory. The dominant mode method yields the stability criteria, which agree well with those from the propagation theory. Therefore, the latter results are compared with available experimental data. It is shown that the initiated instabilities grow until detected experimentally and the upper free interface often behaves similar to a rigid one. 1. Introduction When an initially quiescent, horizontal liquid layer experiences sudden density change by gas absorption into a liquid, the basic density profiles that are driven by mass diffusion develop with time and buoyancy-driven convection can set in at a certain time. For large-Rayeigh-number (large Ra) systems, the critical time to mark the onset of convective motion (tc) becomes an important question, because its prediction is important in analyzing many engineering problems, such as drying of thin paint films, absorption into liquids on packing, and the drawing of polymer filaments from solution. The stability problem in the transient diffusive systems may be called an extended problem of classical Rayleigh-Be´nard convection. The time-dependent instability has been analyzed in heat conduction systems using the frozen-time model,1 propagation theory,2 maximum-Rayleigh-number criterion,3 energy method,4-7 amplification theory,7-9 and a stochastic model.10 The first three models do not require the initial conditions, and these quasisteady-state approximations (QSSAs) yield the critical time as the parameter. The last two models require the initial conditions and the criterion to define manifest convection. In the first, third, and fourth models, the critical conditions are independent of the Prandtl number (Pr). The propagation theory assumes that, at t ) tc, infinitesimal temperature disturbances are propagated, mainly within the thermal penetration depth, and with this length-scaling factor, all the variables and parameters with the length scale are rescaled. The resulting stability criteria have compared relatively well with experimental data in solidification,11 Rayleigh-Be´nard convection,12,13 and Marangoni-Be´nard convection.14 For large Pr values, it has been shown that manifest convection would be first detected at τ ) 4τc. Very recently, Riaz et al.15 used another QSSA model in a porous layer, and they called it the dominant mode method. The QSSA models are based on linear theory. * To whom all correspondence should be addressed. Tel.: +82-64754-3685. Fax: +82-64-755-3670. E-mail:
[email protected].
Here, we will apply Riaz et al.’s dominant mode method15 to the instability problem in an initially quiescent liquid layer that is experiencing a sudden density change due to the gas absorption process. In the present system, with the upper free boundary, the gas is absorbed into the horizontal liquid layer, starting from time t ) 0. For this specific system, the resulting stability criteria will be compared with those from other QSSA models (propagation theory and frozen-time model) and the amplification theory. Predictions will be compared with available experimental data. 2. Theoretical Analysis 2.1. Governing Equations. The system considered here is a Newtonian liquid layer with an initially uniform concentration Ci. For time t g 0, the horizontal layer of liquid depth d experiences absorption through the upper free boundary of interfacial area A and there is no mass transfer through its lower fixed boundary. The schematic diagram of the basic system of mass diffusion is shown in Figure 1. For a high ∆C, buoyancydriven convection will set in after a certain time. Here, ∆C is the concentration difference (∆C ) C* - Ci), with C* ) HPb (where H is the quasi-Henry’s constant and Pb is the partial pressure of absorbing gas in the gas phase). The governing equations of flow and concentration fields then are expressed, using the Boussinesq approximation, as
∇‚U ) 0
(1)
{∂t∂ + U‚∇}U ) - F1 ∇P + ν∇ U + gβCk {∂t∂ + U‚∇}C ) R ∇ C 2
(2)
2
(3)
r
m
where U is the velocity vector, C the concentration, P the dynamic pressure, ν the kinematic viscosity, g the gravitational acceleration constant, F the density, Rm the diffusivity of absorbed gas in liquid, β the concentration coefficient of
10.1021/ie060506b CCC: $33.50 © 2006 American Chemical Society Published on Web 09/09/2006
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Φ(t) ) ≈
Figure 1. Sketch of the basic diffusion state considered here.
expansion, and k the vertical unit vector. The subscript “r” represents the reference state. Here, we will concentrate on the available experimental environments of the sulfur dioxide-water system16 and the carbon dioxide-water system.17 In the fully developed state, the convective motion will disappear, because there is no concentration difference over the liquid layer. The important parameters to describe the present transient system are the Schmidt number (Sc) and the Rayleigh number (Ra), which are defined by the following relationships:
Sc )
ν Rm
and
Ra )
gβ∆Cd3 Rmν
In liquid layers with large Ra values, the critical time to mark the onset of convective motion (tc) becomes an important parameter from the aforementioned equations. For this purpose, we will apply linear theory. For the basic state of diffusion, the dimensionless concentration profile is represented by
∂φ0 ∂φ0 ) 2 ∂τ ∂z
(4)
{Pb - P(t)}V V(Pb - Pi) ) {1 - exp(h*2) efrc(h*)} RT RT
{
V(Pb - Pi) 2 4 h* - h*2 + h*3 + ‚‚‚ RT xπ x 3 π
Φ(t) ≈
2 HA(Pb - Pi)xRmt xπ
(9)
Based on the previous reasoning, eq 6 is approximated as
φ0(τ,z) ) erfc
(2ζ)
(10)
for the case of small h*. This concentration profile is used throughout the present study. However, Tan and Thorpe3 treated the present sulfur dioxide absorption system as a constant flux system. For a constant flux system, the relation Φ(t) ∝ t is more reasonable than Φ(t) ∝ xt. 2.2. Stability Equations. Under linear theory, infinitesimal disturbances that are caused by incipient convective motion at the dimensionless critical time τc can be formulated, in dimensionless form, in terms of the concentration component φ1 and the vertical velocity component w1, by linearizing eqs 1-3:
{Sc1 ∂τ∂ - ∇h }∇h w ) -∇h 2
2
2
1
1
φ1
∂φ0 ∂φ1 + Raw1 )∇ h 2φ 1 ∂τ ∂z
(11) (12)
where the Laplacian is
φ0 ) 0 at τ ) 0
(5a)
dφ0S ∂φ0 V ) at z ) 0 AHRTd dτ ∂z
(5b)
∂φ0 ) 0 at z ) 1 ∂z
(5c)
)
(2ζ + h*)
∇ h2 )
∂2 ∂2 ∂2 + 2+ 2 2 ∂x ∂y ∂z
and the horizontal one is
∇ h 12 )
where V is the volume of the gas phase, R the gas constant, and T the temperature. In the aforementioned equations, τ ) Rmt/d2, z ) Z/d, and φ0 ) (C - Ci)/(C* - Ci). For deep-pool systems of τ e 0.01, the Leveque-type solution is given as
φ0 ) exp(h*ζ + h*2) erfc
(8)
Note that the first term in the aforementioned expansion is the exact solution for the case of absorption with a constant surface concentration. For small h*, the first term is dominant and the previously described relation can be approximated as
with the following initial and boundary conditions:
(
}
(6)
where h* ) AHRTdxτ/V and ζ ) z/xτ. The upper surface concentration φ0S can be obtained as
∂2 ∂2 + ∂y2 ∂y2
Here, the velocity component has the scale of Rm/d and the temperature component has that of Rmν/(gβd3). The proper freerigid boundary conditions are given by
w1 ) w1 )
∂2w1
) φ1 ) 0 at z ) 0
(13a)
∂w1 ∂φ1 ) ) 0 at z ) 1 ∂z ∂z
(13b)
∂z2
(7)
For a given Sc value and Ra value, the critical time τc should be determined using eqs 11-13. Now, convective motion is assumed to exhibit the horizontal periodicity and the normal-mode analysis is used. The perturbed quantities then can be expressed as
During the diffusion period, the total amount of gas absorbed up to time t, Φ(t), is
[w1(τ,x,y,z),φ1(τ,x,y,z)] ) [w*1 (τ,z),φ*1 (τ,z)] exp[στ + i(ax x + ay y)] (14)
φ0S )
CS - Ci P(t) - Pi ) ) exp(h*2) erfc(h*) C* - Ci P b - Pi
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where “i” is the imaginary number and the horizontal wavenumber a has the relation of a ) [ax2 + ay2]1/2. Now we will apply the QSSA models: (i) the dominant mode method, (ii) propagation theory, and (iii) the frozen-time model. Riaz et al.15 called their model the dominant mode method, and they applied it to the porous layer. These three models may be called dominant mode analysis, because the earliest time τc and its corresponding wavenumber ac should be determined to satisfy eqs 11-14 for a given Ra value and Sc value. Their solution procedure is well-known.15,18 We neglect the first term that involves Sc in eq 11, because the Sc value is on the order of 1000 in the present absorption system. 2.3. Dominant Mode Method. For deep-pool systems of small time, it may be natural that φ*1(τ,z) ) φ*(τ,ζ) and w*1(τ,z) ) w*(τ,ζ), considering eq 10. We transform the disturbance equations such that the eigenfunctions that are associated with the diffusion operator are localized around the base-concentration front. Now, eqs 11-13 are transformed to the similarity variable of the base state, ζ ) z/xτ. The perturbation equations then can be expressed as
(
)
2 ∂2 2 a τ w* ) -a2τφ* for Sc f ∞ 2 ∂ζ
(
and
Lφ1 ) -φ1(ζ)
The long-wave mode solution shows that the first mode of the self-similar diffusion operator decays with time, whereas the remainder of the spectrum decays more rapidly. Therefore, the first mode is the dominant one. Using the dominant mode solution φ* ) A1(τ)ζ exp(-ζ2/4), substituting it into eq 16, and integrating across the domain, we obtain
τ
dA1 dθ0 dA1 ) στA1 ) ) -(1 + a2τ)A1 + Raτ3/2w* dτ d ln τ dζ (22a)
where
〈 σ)
)
∂2w* ) φ* ) 0 at ζ ) 0 ∂ζ2
(17a)
w* )
∂w* ∂φ* ) ) 0 as ζ f ∞ ∂ζ ∂ζ
(17b)
The self-similarity applies to the base-concentration field (see eq 10), but the amplitudes have a time-dependent spatial structure. By following Riaz et al.’s procedure,15 the streamwise operator of the concentration disturbance in the transformed coordinate, ζ, is expressed as
∑ Ak(τ)φk(ζ) k)1
(
ζ ζ H 4 k2
( ) ζ2 4
(22c)
2
( )
ζ2 w* ) -a2τζ exp A1 4
(23a)
w*(τ,ζ) can be obtained analytically as
( ) [
1 w* ) {f1(ζ) - f1(∞)}ζ exp(axτζ) + A1 4 {f2(ζ) - f1(∞)}ζ exp(-axτζ) {p1(ζ) - p1(∞)} exp(axτζ) {p2(ζ) + p1(∞)} exp(-axτζ) -
{f1(ζ) - f1(∞)} exp(axτζ)
{f2(ζ) - f1(∞)} exp(-axτζ)
+
(
axτ
]
(23b)
x dx )
exp(a2τ) f2(ζ) )
(
)
2
∫0ζ exp axτx - x4
(for k ) 1, 2, 3, ...) (20)
(21a)
)
2
∫0ζ exp -axτx - x4
(19)
The eigenfunctions φk of L are the Hermite polynomials Hk(ζ/2) in the semi-infinite domain with the weight function exp (-ζ2/4). The eigenvalues are λk ) - (k + 1)/2 for k ) 1, 2, ... The most unstable eigenfunction, i.e., the dominant mode solution that satisfies the aforementioned boundary conditions in eqs 17, is given by
φ1 ) ζ exp -
)( )
∂2 - a2τ ∂ζ2
f1(ζ) )
with
( ) ()
〈 〉
where
∞
Lφk ) λkφk(ζ) ) λk exp -
( )
1 w*dθ0 1 dA1 ) - + a2 + Raτ3/2 A1 dτ τ A1 dζ
(18)
and the temperature disturbance is expanded as
2
(22b)
∫0∞ζ exp(-ζ2/4) dζ
axτ
ζ ∂ ∂2 L) 2+ 2 ∂ζ ∂ζ
φ*(τ,ζ) )
〉
∫0∞w* dζ0 dζ
By solving the equation
with the following conditions for the upper free and the lower rigid boundaries:
w* )
dθ
dθ0 w* ) dζ
(15)
dφ0 ζ ∂ ∂θ* 1 ∂2 + - a2τ φ* ) Raw*xτ (16) ∂τ τ ∂ζ2 2 ∂ζ dζ
(21b)
2
x dx )
exp(-a2τ) p1(ζ) )
∫0ζ exp{-(axτ + 2x) }x dx
∫0ζ exp
(
{(
- -axτ +
)
)}
x2 x dx 2
2
∫0ζxf1′ dx ) ∫0ζ x2 exp -axτx - x4
dx
and
p2(ζ) )
(
)
2
∫0ζ xf2′ dx ) ∫0ζx2 exp axτx - x4
dx
The driving force for the instability, the integral 〈w*(dφ0/dζ)〉, is also a function of time and the wavenumber. For a given Ra
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(
)
∂C0 Rmν Rmν ∆C ∆C ) ) ∼ Ra∆-1 4 3 ∂Z ∆ ∆ gβ∆ gβ∆C∆
(28)
where Ra∆ is the Rayleigh number based on the penetration depth ∆. With increasing Ra, both the onset time tc and its corresponding thickness ∆ become smaller and the characteristic value of Ra(∆/d)3 (i.e., Raτ3/2) will become a constant. To force the self-similar transformation, we assume dimensionless amplitude functions of disturbances, based on eq 27:
w*1 ) τw*(ζ)
(29a)
φ*1 ) φ*(ζ)
(29b)
and
Figure 2. Marginal stability curves for deep-pool systems of Sc f ∞.
value, the critical conditions can be determined analytically from eq 22 when
∂τ )0 ∂a
(24)
under the condition of σ ) 0 with the critical wavenumber ac. The resulting τ value is the minimum, i.e., τc. With the dominant mode method, the marginal stability curve for the present absorption system of small time is given in Figure 2 and the critical conditions are obtained as follows:
Raτc3/2 ) 8.08
This satisfies the stability criteria suggested by Park et al.:18
r 0 ) r1
(30)
where r0 denotes the temporal growth rate of the base concentration and r1 is its perturbed growth rate. They are defined as the root-mean-square quantities of the concentration components:
r0 )
1 d〈φ0〉
(31a)
〈φ0〉 dτ
and
r1 )
(25a)
1 d〈φ1〉
(31b)
〈φ1〉 dτ
where
and
acτc1/2 ) 0.33
(25b)
al.15
claimed that this dominant mode method would Riaz et yield exact results for small times but deviates from the exact solution for large times. 2.4. Propagation Theory. The propagation theory used to find the onset time of convective motion (i.e., the critical time tc) is based on the assumption that, in deep-pool systems, the infinitesimal concentration disturbances are propagated mainly within the solutal penetration depth at the onset of convective motion ∆ (∆ ∝ xRmt) and the following scale relations are valid for perturbed quantities from eqs 2 and 3,
ν W1
W1 ∆2
∼ gβC1
∂C0 C1 ∼ Rm 2 ∂Z ∆
(26a)
where δ is the usual dimensionless solutal penetration depth (δ ) ∆/d ∝ xτ). The relation given in expression 26b yields
A
(D2 - a*2)2w* ) a*2φ*
(32)
(D + 21ζD - a* )φ* ) Ra*w*Dφ* 2
(27)
x
∫A(quantity)2 dA)
and dA ) S dz with S ) πd/ac. It is interesting that the dominant mode solution, φ* ) A1(τ)ζ exp(-ζ2/4), also satisfies the aforementioned relation of r0 ) r1 at the marginal condition of σ ) 0 in eq 22. The propagation theory is a QSSA model in the localized (τ, ζ) domain, rather than the global (τ, z) domain, and satisfies the marginal stability condition of ∂φ*/∂τ ) 0 in the former domain. Its amplitude function of concentration disturbances is a function of ζ only in the localized domain: ∂φ*1 /∂τ ) -(ζ/ (2τ)) dφ*/dζ and ∂φ*/∂τ ) 0. Using the previously described reasoning, we introduce the self-similar transformation with D(‚) ) ∂(‚)/∂ζ and, for the present system of very large Sc values, we obtain the stability equations from eqs 11-13:
(26b)
from the balance between viscous and buoyancy terms in eq 2 and also from that among terms in eq 3. Now, based on the relation given in expression 26a, the following amplitude relation is obtained in dimensionless form:
w*1 ∼ δ2 ∼ τ φ*1
〈quantity〉 )
(
2
0
(33)
The aforementioned equations can be produced from eqs 15, 16, and 29, under the localized (τ,ζ) domain. The proper boundary conditions are given as follows:
w* ) D2w* ) φ* ) 0 at ζ ) 0
(34a)
w* ) Dw* ) Dφ* ) 0 as ζ f ∞
(34b)
where a* ) a xτ. For a given Sc value, Ra* and a* have been treated as eigenvalues. For small times, the modified Rayleigh
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good results for large times, but it is ill-suited for the task of resolving the small-time behavior. Homsy5 and Neitzel6 analyzed the stability problem under the impulsively changing temperature profile through application of the energy method. Because of its poor convergence and computational load for the large-Ra system, it is too difficult to obtain the exact global stability limit for small times. Based on Neitzel’s calculation results6 (his Figure 2) and our calculation results, the global stability limit at small times shows the relation of Raτc3/2 < 1, similar to that of the frozen-time model. 2.6. Comparison among Marginal States for Sc f ∞. The marginal stability curves from the aforementioned three QSSA models are compared in Figure 2, and the resulting normalized amplitude functions of w* and φ* are shown in Figure 3. The results from the propagation theory and dominant mode method satisfy the following relation: Figure 3. Amplitude profiles of disturbances at τ ) τc for deep-pool systems.
number Ra* (defined as Ra* ) Raτ3/2) has been used in stability analyses.4,10,14 Now, the minimum value of Ra* should be determined in the plot of Ra* vs a*, under the principle of an exchange of stabilities. In other words, the minimum value of τ (i.e., τc) and its corresponding wavenumber (ac) are obtained for a given Ra. This type of treatment was introduced by Ball and Himmelblau.19 Now, it is known that Riaz et al.’s dominant mode method15 is a good approximation of the propagation theory. The former method would not yield the exact solution, and its present solution method can be applied to the limiting case of large Sc values. Therefore, we believe that the latter model is a better one. The critical conditions predicted by the propagation theory are
Raτc3/2 ) 8.63
(35a)
and
acτc1/2 ) 0.30
r 0 ) r1 )
(D2 - a*2)2w* ) -a*2θ* and
(D2 - a*2)θ* ) Ra*w*Dθ0 instead of eqs 32 and 33. The resulting stability criteria become independent of Sc, and τc is obtained easily for a given Ra. The resulting neutral stability curve is shown in Figure 2. The resulting τc values are known to be too small, in comparison with those from other models. The frozen-time model yields
(36)
which we have suggested (see eq 30). However, the frozentime model yields the relation of r0 ) 0 < r1 for small times. Figure 3 shows that, with the former two models, φ* is propagated mainly within the solutal penetration depth of diffusion. 3. Results and Discussion Because gas absorption systems involve a large Sc value, we are concerned with the predictions for the case of Sc f ∞. Based on the stability criteria from the propagation theory, the critical conditions to mark the onset of convective instability are expressed as follows:
tc,F ) 4.21
(
)
-2/3
gβ∆CRm1/2 ν
and
( )
gβ∆C 2π ) 0.149 λc,F Rmν
(35b)
which are the same from the theory of Park et al.20 and that of Ihle and Nin˜o.21 They applied the conditions of the isothermal, rigid boundary in the heat conduction system; however, their predictions for small times can be applied to the present absorption system, because the boundary condition of ζ f ∞ has little effect on the critical conditions. The aforementioned critical values are similar to the results from the dominant mode method (see eq 25), as shown in the comparison given in Figure 2. 2.5. Frozen-Time Model. Under the QSSA, the disturbance equations (eqs 11-13) can be solved in a straightforward manner, using the frozen-time model.1 The conventional frozentime model neglects the terms that involve ∂(‚)/∂τ in eqs 11 and 12 in the global (τ,z) domain. This results in
1 as τ f 0 4τ
1/3
for large Ra (37)
where λ is the wavelength and the subscript F denotes the upper free boundary. In gas absorption into water, the upper free boundary often behaves similar to the rigid one. This peculiar behavior will be discussed later. For this case of rigid-rigid boundaries, Kim et al.’s22 predictions yield the following:
(
)
gβ∆CRm1/2 tc,R ) 7.38 ν
-2/3
and
( )
gβ∆C 2π ) 0.195 λc,R Rmν
1/3
for large Ra (38)
where the subscript R denotes the upper rigid boundary. As expected, tc and λc decreases with an increase in ∆C. For a given Rayleigh number, the fastest growing mode of infinitesimal disturbances would set in at t ) tc with a ) ac, i.e., λc. This instability will grow and it will be observed at t ) to. Experimental results show that the initiated cell structure is quite well preserved, even after the convective motion becomes quite pronounced.9 It seems evident that the cell size is almost constant during the period of tc e t e to, but its vertical growth will be continued. This type of behavior is illustrated in the numerical simulation of Park and co-workers.18,20 Foster’s8 amplification theory concerns the detection time of convective motion (to), where the white noise is usually assumed
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Figure 4. Comparison of predictions with experimental to values for water systems. Experimental to data are from Blair and Quinn16 and Plevan and Quinn,27 and all physical properties that have been suggested in the former work are used.
Figure 5. Effect of boundary conditions on the onset time of convective motion for sulfur dioxide absorption into water. Experimental to data shown by squares are from Blair and Quinn,16 and all physical properties suggested in their work are used.
as the initial condition. Using this model, Mahler and Schechter9 analyzed the present instability problem. They expanded disturbance quantities into a series of orthonormal functions that satisfy the boundary conditions automatically. This model defines the amplification ratio w j as
w j)
(
)
∫01 w*1 2(τ,z) dz 1/2 ∫01 w*1 2(0,z) dz
(39)
They assumed that manifest convection would be detected first at the characteristic time τ0, when the w j value reaches 1000. Their predictions are given for Sc ) 600:
Raτ0,F3/2 = 110 and aτc,F1/2 = 0.43 for free-rigid boundaries (40) Raτ0,R3/2 = 290 and aτc,R1/2 = 0.73 for rigid-rigid boundaries (41) Note that the constants in eq 40 is much larger than those in eq 35, and the former constants are dependent on the initial conditions and the amplification ratio. Foster23 commented that, with the correct dimensional relationships, the result of to = 4tc would be maintained. This means that the fastest growing mode of instabilities, which sets in at t ) tc, will grow with time until manifest convection is detected near the entire upper boundary at t = 4tc. The validity of to = 4tc requires further study, but this relationship is observed, with the propagation theory, in various transient heat and momentum diffusive systems2,22,24-26 and even in eqs 35 and 40. Therefore, based on tc values predicted from the propagation theory (i.e., eqs 37 and 38), we use the relation of
to ) 4tc
(42)
The aforementioned characteristic times are compared with available experimental data of gas absorption into water in Figure 4. It is known that convective motion (or deviation from eq 9) is detected first at t ≈ 4tc,R, even in the free boundary system. To examine this strange behavior on the gas/liquid interface, Blair and Quinn16 covered the upper surface with fine steel wire cloth and conducted absorption experiments with sulfur dioxide. Mahler and Schechter9 conducted experiments through surfaces covered by various surfactants. However, there
Figure 6. Effect of surfactant on the onset time of buoyancy-driven convection for ethyl ether systems. Experimental to data are from Blair and Quinn,16 and all physical properties suggested in their work are used.
was no significant difference between the rigid-rigid and the free-rigid systems. These results are compared with the aforementioned predictions in Figure 5. Scriven and Sternling28 mentioned that the rigidlike behavior of the upper free boundary of the water surface may be related to surface elasticity and viscosity. Figure 6 shows that the relation of to ≈ 4tc,R is maintained; even the desorbing of ethyl ether from monochlorobenzene covered with trace amounts of surfactant materials can make the interface rigid, but with a pure solution, other modes of instability (such as surface-tension-gradient-driven convection) are possible. Similar experiments were conducted in an aqueous carboxy methyl cellulose (CMC) solution.29 For the systems of sulfur dioxide absorption into CMC solution, the detection times (to) are compared in Figures 7 and 8. There is no significant difference between CMC solution and water experiments. However, for the case of carbon dioxide, the convective motion is detected at t = 4tc,F, as shown in Figure 8. Interestingly, in carbon dioxide, absorption into CMC solution follows the relation of eq 42. To explain these interesting phenomena in gas absorption/desorption systems, Davenport and King30 mentioned that the major reason for the retardation of the detection time of convective motion might be the difference in the initiation mechanism, which is affected by the diffusion length: xRmto in mass diffusion systems and xRto in heat
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Figure 7. Comparison of characteristic times for sulfur dioxide-water and sulfur dioxide-carboxy methyl cellulose (carbon dioxide-CMC) solution systems. Experimental to data are from Tan and Thorpe.29
Figure 8. Comparison of characteristic times for carbon dioxide-water and carbon dioxide-CMC solution systems. Experimental to data are from Tan and Thorpe.29
conduction systems. Here, R is the thermal diffusivity. Usually, the former length scale is much smaller than the latter one, and, therefore, there is a possibility that the initiation mechanism is different over the range of diffusion thickness. According to their suggestion, the plot of to/tc,F vs xRmto is given in Figure 9. This figure shows that, if the diffusion length at the detection time is smaller than 4 × 10-1 mm, the upper free interface behaves similar to a rigid boundary. However, its justification requires further study. The cell size is compared with experimental data of the sulfur dioxide-water system in Figure 10. The average cell sizes are similar to the λF values obtained from eq 38. The widely scattered experimental data reflect the experimental difficulties. Park et al.20 showed that convective motion is very weak during the period of tc e t e to, and during this period, the related mass transport is well-represented by the diffusion state. Their numerical simulation shows smaller tc values than those from eq 35 and encounters difficulties in choosing the initial conditions. A more refined study is required for understanding properties of characteristic times and free boundary conditions in gas absorption processes. 4. Conclusions The critical condition to mark the onset of convective motion in gas absorption systems has been analyzed using the dominant
Figure 9. Plot of to/tc vs xRmto. Experimental to data are from Blair and Quinn,16 Plevan and Quinn,27 and Tan and Thorpe.29
Figure 10. Comparison of predicted critical wavenumbers with Blair and Quinn’s16 experimental data.
mode method and the frozen-time model. The resulting stability criteria are compared with the existing results from the propagation theory and amplification theory. It is shown that, for large Ra, the dominant method approximates the propagation theory well. Based on the latter model, predictions are compared with available experimental data. The present predictions on the detection time of manifest convection show the relation of to = 4tc,R for smaller diffusion thickness, and a relationship of to = 4tc,F is observed for larger diffusion thicknesses. However, its further justification is required. It is concluded that the propagation theory can be used to predict the detection time of convective motion in simple diffusive systems, but more-refined work is required theoretically and also experimentally. Notations A ) interfacial area (m2) a ) dimensionless horizontal wavenumber; a ) a* ) modified wavenumber; a* ) aτ1/2 C ) concentration in liquid (kg/m3) d ) depth of liquid layer (m) D ) differential operator; D ) d/dζ g ) gravitational acceleration (m/s2) H ) quasi-Henry’s constant (kg-mol/(kg Pa)) P ) pressure (Pa) R ) gas constant (Pa m3/(kg-mol K)) Ra ) Rayleigh number; Ra ) gβ∆Cd3/Rmν
xax2+ay2
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Ra* ) modified Rayleigh number; Ra* ) Raτ3/2 Sc ) Schmidt number; Sc ) ν/Rm T ) temperature (K) t ) time (s) U ) velocity vector (m/s) V ) volume of gas phase (m3) W ) vertical velocity (m/s) w ) dimensionless vertical velocity x, y, z ) dimensionless Cartesian coordinates Greek Letters R ) thermal diffusivity (m2/s) Rm ) mass diffusivity (m2/s) β ) volumetric solutal expansion coefficient (1/(kg-mol/m3)) ∆C ) concentration difference (kg-mol/m3) ∆ ) penetration depth (m) δ ) dimensionless penetration depth; δ ) ∆/d ζ ) similarity variable; ζ ) z/xτ λ ) wavelength (m) µ ) viscosity (Pa s) ν ) kinematic viscosity (m2/s) F ) density (kg/m3) τ ) dimensionless time; τ ) Rmt/d2 φ1 ) dimensionless perturbed concentration; φ1 ) C1g(- β)d3/(Rmν) φ0 ) dimensionless basic concentration; φ0 ) (C - Ci)/∆C Subscripts b ) bulk phase c ) critical state F ) free boundary system i ) initial state o ) observable condition R ) rigid boundary system s ) surface condition x ) x-direction y ) y-direction 0 ) basic state 1 ) perturbed state Superscript * ) transformed quantity Acknowledgment This work was supported partially by LG Chemical, Ltd., Seoul, under the Brain Korea 21 Project of the Ministry of Education. Literature Cited (1) Kurenkova, N.; Eckert, K.; Zienicke, E.; Thess, A. Desorption-driven convection in aqueous alcohol solution. Lect. Notes Phys. 2003, 628, 403. (2) Kim, M. C.; Park, J. H.; Choi, C. K. Onset of buoyancy-driven convection in the horizontal fluid layer subjected to ramp heating from below. Chem. Eng. Sci. 2005, 60, 5363. (3) Tan, K.-K.; Thorpe, R. B. The onset of convection induced by buoyancy during gas diffusion in deep fluids. Chem. Eng. Sci. 1999, 54, 4179. (4) Wankat, P. C.; Homsy, G. M. Lower bounds for the onset time of instability in heated layers. Phys. Fluids 1977, 20, 1200.
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ReceiVed for reView April 21, 2006 ReVised manuscript receiVed August 5, 2006 Accepted August 15, 2006 IE060506B
