4 Review of the treatment of chemical kinetics
4.3 Microkinetics of dispersed systems
4.3.2 Mass transfer in the dispersed phase
Table 16 (p. 117) shows a compilation of correlations for mass transfer in the dispersed phase. The correlations are represented in terms of Sh. The properties of the continuous and dispersed phases are indicated with subscripts c and d, respectively. Under conditions of negligible external resistance the value of the Sherwood number is defined by [288]:
Sh= 2
3(1−F) dF dFoM = 2
3Φ
d(1−Φ)
dFoM . (134)
1 E - 0 2 1 E - 0 1 1 E + 0 0 1 E + 0 1 1 E + 0 2 1 E + 0 3
Fig 16. Shape regimes for bubbles and droplets in unhindered gravitational mo-tion through liquids. Modified after [288].
The time-averaged Sherwood number is then obtained from [288, 315]:
Sh=−2ln(1−F)
3FoM =2lnΦ
3FoM. (135)
In a stagnant sphere (Re = 0, PeM= 0) mass transfer takes place solely by molecular diffusion [1, 288, 322]. The concentration change with respect to time is described by the Fick’s 2nd law for spherical coordinates [1]:21
∂c
wherexdenotes the distance from the midpoint of the sphere. A series expansion for the solution of the average concentration from Eq. 136 is known as the Newman [323]
solution, which is expressed in terms of Sh in Eq. 141 (Table 16). The steady-state asymptotic value of the Newman solution is [1, 288, 324]:
FolimM→∞Sh=2π2
3 ≈6.58. (137)
21Here, the mass diffusivity (D) is taken as a constant in space.
If the dispersed phase is in motion relative to the continuous phase, the external flow can induce internal circulation or oscillation, which decrease the dispersed mass transfer resistance considerably [1, 116, 288, 322, 325, 326]. The characteristic time variation of the Sherwood number in the dispersed phase in a creeping flow is illustrated in Fig. 17.
The time variations are caused by Hill’s vortex [327], whose intensity is controlled by the continuity of the velocity and the viscous shear stress at the interface [324].
0 . 0 0 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 0
5
1 0 1 5 2 0 2 5 3 0 3 5
P e M 1 + d/c ↓
4 0 0 1 0 0 0
1 6 0 8 0
∞
Sherwood number
F o u r i e r n u m b e r f o r m a s s t r a n s f e r
0
Fig 17. Variation of instantaneous overall Sh number in a creeping flow with neg-ligible external resistance. Redrawn after [288].
Kronig and Brink [328] proposed a mathematical solution, which describes laminar diffusion with circulation induced by viscous forces (see Eq. 144 in Table 16). The first seven values for the parametersAiandλiin Eq. 144 are shown in Table 15.
Table 15. Parameters of the Kronig and Brink solution. Adapted from [329].
i
Parameter 1 2 3 4 5 6 7
Ai 1.33 0.60 0.36 0.35 0.28 0.22 0.16
λi 1.678 8.48 21.10 38.5 63.0 89.8 123.8
The Kronig-Brink solution is based on the assumption that surfaces of uniform concentration coincide with the Hadamard-Rybczynski streamlines [330, 331], which
describe the laminar flow past a fluid sphere in the Stokes flow regime (Re1) [288, 325, 326, 328]. In terms of dimensionless numbers, the solution is thus applicable for the special case of Re→0 and PeM→∞[288, 324]. Despite its limited theoretical applicability, experimental studies have shown that the Kronig-Brink solution gives a reasonably good prediction of the mass transfer coefficient even at Reynolds numbers well above those corresponding to creeping flows [288, 325, 332]; the resulting error is then O(Pe−0.5) [333]. Johns and Beckmann [334] have suggested that for practical purposes the Hadamard stream function may be extended to systems for which Re < 10.
The steady-state asymptotic value of the Kronig-Brink solution is given by [324]:
FolimM→∞Sh=32
3 λ1≈17.9, (138)
which indicates that laminar circulation should increase the internal mass transfer rate by a factor of 2.7 in comparison to rigid sphere behaviour. In the case of heat transfer within a sphere, the asymptotic value of Nu is within 5% of the Newman solution for PeH/(1−µµdc)<10 and within 5% of the Kronig-Brink solution for PeH/(1−µµdc)>250 [288, 326]. Fig. 18 shows the numerical results by Juncu [326] in comparison to the Newman and Kronig-Brink solutions.
A somewhat simpler expression for mass transfer in a droplet with internal circulation was proposed by Calderbank and Korchinski [335]. Employing the approximate Vermeulen [336] empirical approach (Eq. 143 in Table 16) as the basis of their analysis, they considered the effect of internal circulation on effective diffusivity by multiplying the value of molecular diffusivity (Eq. 142 in Table 16). The steady-state asymptotic value of their correlation is
FolimM→∞Sh=3π2
2 ≈14.8, (139)
which suggests that the effective mass diffusivity during internal circulation is 2.25 times the molecular value [335]. It should be noted that neither the Kronig-Brink nor the Calderbank-Korchinski solution reproduce the time oscillations shown in Fig. 17.
It is well known that internal circulation can be inhibited or even suppressed by surface active elements [1, 288, 325, 337–339], and that the tendency for this phenomenon grows with increasing droplet size [1, 338]. In a moving bubble or drop the surfactants are swept to the aft and the resulting concentration gradient induces a surface tension gradient and a tangential stress, which opposes the direction of movement [339].
Henschke and Pfennig [340] showed that the effect of surface instability on the internal
1 1 0 1 0 0 1 0 0 0 1 0 0 0 0
05
1 0 1 5 2 0 2 5 3 0
K r o n i g - B r i n k s o l u t i o n J u n c u
R ei n t < 1 R ei n t = 1 R ei n t = 1 0 R ei n t = 1 0 0
Asymptotic Nusselt number
P é c l e t n u m b e r f o r h e a t t r a n s f e r ( R e i n tP r ) N e w m a n s o l u t i o n
Fig 18. Asymptotic Nu values for different PeHand Reintvalues in a liquid–liquid system. Comparison of numerical results of Juncu [326] with the Newman [323]
and Kronig-Brink [328] solutions.
circulation can be described by the Newman solution by replacing the mass diffusivity with effective diffusivity:
Deff=D+ udp
CIP
1+µµdc, (140)
whereCIPis a substance specific constant which accounts for transient effects at the interface, whileµdandµcdenote the dynamic viscosity of the dispersed and continuous phases, respectively. The corresponding correlation for the Sherwood number is shown in Eq. 145 (Table 16). Later it has been shown thatCIPis not only substance specific, but also a function of concentration [316].
A mathematical solution for internal circulation at larger Re values was proposed by Handlos and Baron [341] (Eq. 146 in Table 16), who assumed eddy diffusion between internal toroidal lines of a moving drop. The solution assumes a constant spherical shape of the drop [315], and suggests that Sh is a function of ReSc,i.e.PeM, and the viscosity ratio of the dispersed and continuous phases. The value ofλ1in Eq. 146 was reported to be 2.88 [341], but was later more accurately recalculated to be 2.866 by Wellek and Skelland [342]. Because only the first term in the series solution was used,
the Handlos-Baron solution is inaccurate at small contact times [315, 343, 344].
Olander [343] modified the Handlos-Baron solution to be applicable for the entire range of contact times (Eq. 147 in Table 16). Another modification of the Handlos-Baron solution was proposed by Wegener and Paschedag [345] (Eq. 148 in Table 16), who introduced a concentration dependent parameterαto describe the influence of the initial solute concentration on Marangoni convection. However, as the internal mass transfer is usually not the rate limiting step at high PeMvalues, mass transfer correlations of the type shown in Eqs. 146, 147, and 148 have found little use in the metallurgical literature.
Table16.Masstransfercorrelationsfordispersedphase. ReferenceApplicationEquation Newman[323]Moleculardiffusioninastagnantdrop (Re=0,PeM=0)Sh=2π2 3
∞ ∑ i=1 exp−i2π2FoM ∞ ∑ i=1h 1 i2exp(−i2π2FoM)i(141) CalderbankandKorchinski[335]EffectivemoleculardiffusioninadropSh=3π2 4
exp −9π2 4FoM F(1−F)(142) whereF= 1−exp −9π2 4FoM1/2 (143) KronigandBrink[328]Laminardiffusionwithcirculationinducedby viscousforcesbetweenamovingdropand continuousphase(Re→0,PeM→∞)
Sh=32 3
∞ ∑ i=1 A2 iλiexp(−16λiFoM) ∞ ∑ i=1 A2 iexp(−16λiFoM)(144) HenschkeandPfennig[340]Effectivemasstransferinadropinthepres- enceoftransientinstabilitySh=2π2 3
∞ ∑ i=1 exp−i2π2Fo0 M ∞ ∑ i=1h 1 i2exp(−i2π2Fo0 M)i(145) Notes:Eqs.141,142,and144aretakenfrom[288],[346],and[324],respectively;F=x0−x x0−x∗=fractionalequilibrium.Fo0 M=4Defft d2 p=modifiedFoM.
Table16.(Continued) ReferenceApplicationEquation HandlosandBaron[341]Eddydiffusionbetweeninternaltoroidal streamlinesofamovingdrop
Sh=λ1 128ReSc 1+µd µc(146) Olander[343]Eddydiffusionbetweeninternaltoroidal streamlinesofamovingdropSh=0.972λ1 128ReSc 1+µd µc! +0.3 FoM(147) WegenerandPaschedag[345]Eddydiffusionbetweeninternaltoroidal streamlinesofamovingdropinthepresence ofMarangoniconvection Sh=αλ1 128ReSc 1+µd µc
! (148)