Homogeneous–heterogeneous regime transition in bubble columns

Chemical Engineering Science 56 (2001) 4609–4626

www.elsevier.com/locate/ces

Homogeneous–heterogeneous regime transition in bubble columns M. C. Ruzickaa;∗ , J. Zahradn*+ka , J. Draho-sa , N. H. Thomasb a Institute

of Chemical Process Fundamentals, Czech Academy of Sciences, Rozvojova 135, 16502 Prague, Czech Republic of Chemical Engineering and Applied Chemistry, Aston University, Aston Triangle, Birmingham B4 7ET, UK

b Department

Received 16 November 1999; received in revised form 26 February 2001; accepted 26 March 2001

Abstract A simple physical model for homogeneous–heterogeneous regime transition in bubble columns is developed. The model is based on hydrodynamic coupling between gas and liquid phases. For the homogeneous regime, the coupling is made via bubble drift concept (Darwin, Proc. Camb. Phil. Soc. 49 (1953) 342). As a byproduct, a novel non-empirical formula for bubble slip velocity results, u=w = 1 − ae=(1 − e). For the heterogeneous regime, the coupling is obtained in a simple formal way, recovering the classical result of Zuber and Findlay. The regime transition is considered as a smooth and gradual process characterized by a transition function. The model has ;ve parameters: two terminal bubble velocities, bubble drift coe=cient, Zuber–Findlay constant, and intermittency factor. All have clear physical meaning and are extractable from experimental data. The model gives formulas for the voidage-gas ?ow rate dependence separately for the homogeneous regime, heterogeneous regime and transition regime. The model gives a kinematic stability condition for the homogeneous regime and predicts the critical gas ?ow rate where the transition begins. It also predicts the maximum possible gas holdup in bubble columns. The model is veri;ed by experiments with four diAerent air–water bubble columns. Good agreement is found. Our results are compared with results of other authors. The model agrees with the drift-?ux concept developed by Wallis (One-dimensional Two-phase, Flow, McGraw-Hill, New York, 1969) and with the stability theory of Shnip et al. (Int. J. Multiphase Flow 18 (1992) 705). ? 2001 Elsevier Science Ltd. All rights reserved. Keywords: Bubble column; Flow regime; Stability; Drift; Slip velocity

1. Introduction Two basic ?ow regimes occur in bubble columns, the homogeneous and the heterogeneous ones (Deckwer, 1992; Kastanek, Zahradnik, Kratochvil, & Cermak, 1993; Molerus, 1993; Zahradnik et al., 1997). Their typical features are shown in Fig. 1 and their diAerences are listed in Table 1. The homogeneous regime, HoR, (also: laminar, uniform, dispersed, bubbly ?ow regime) is produced by plates with small and closely spaced ori;ces (plate type I) at low gas ?ow. Uniform layers of equal-sized densely packed non-coalescing small and almost spherical bubbles are generated continuously at the plate. The bubbles rise almost vertically and lift up a considerable amount of liquid to the top of the column. The liquid thus carried up must return down, as there is zero net liquid ?ow in the ∗ Corresponding author. Tel.: +420-2-2039-0299; fax: +420-22092-0661. E-mail address: [email protected] (M. C. Ruzicka).

column. The liquid counter-current delays the bubble rise, hence increasing the gas holdup. This small-scale liquid velocity ;eld is unsteady and highly ?uctuating on short time scales; however, the long-time radial pro;les of velocity (Hills, 1974; Lapin & Lubbert, 1994) and voidage (Kumar, Moslemian, & Dudukovic, 1997) are ?at. The heterogeneous regime, HeR, (also: turbulent, circulation, clustered, churn-turbulent regime) is produced by either (i) plates with small and closely spaced ori;ces (plate type I) at high gas ?ow, or, (ii) plates with large ori;ces (plate type II) at any gas ?ow. The former case results from the instability of HoR and the subsequent transition, and will be denoted as THeR. The latter case results from non-uniform gas distribution at the plate due to large ori;ces and their large spacing, and will be called ‘pure heterogeneous regime’, PHeR. Regardless of their diAerent origins, THeR and PHeR are almost undistinguishable: populations of large and highly non-uniform bubbles with a strong tendency to coalesce are generated in both cases. Therefore, PHeR serves as a suitable reference

0009-2509/01/$ - see front matter ? 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 1 ) 0 0 1 1 6 - 6

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Fig. 1. Typical features of homogeneous and heterogeneous ?ow regimes in bubble column. (a) Instantaneous ?ow patterns. (b) Time-averaged radial pro;les. (c) Voidage-gas ?ow rate dependence e(q).

state for indicating the homogeneous regime and its transition. Indeed, carrying out experiments in two columns diAering only in the plate type, both e–q graphs start at zero, then split, and then rejoin again at a high gas ?ow when the transition is completed. The liquid ?ow in HeR is substantially diAerent from that in HoR. Large-scale and strong non-uniformities in buoyancy distribution at the plate trigger and drive large-scale and strong convec-

tive motions of liquid within the whole column—liquid circulations. High-voidage regions are accelerated and advected to the top where bubbles escape at the surface, and the bubble-free liquid ?ows back down. Thus the upward liquid motion enhances the bubble rise, hence decreasing the gas holdup. The liquid circulations are highly non-stationary on short time scales (Chen, Reese, & Fan, 1994; Devanathan, Dudukovic, Lapin, & Lubbert, 1995),

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Table 1 Qualitative comparison of homogeneous and heterogeneous ?ow regimes in bubble columns Flow regime Homogeneous

Heterogeneous

Plates

Ori;ce size Ori;ce pitch Number of ori;ces

Small Small Large

Large Large Small

Bubbles

Size Formation Coalescence Rise

Small Break-up of ;ne jets No Almost vertically

Large Break-up of strong jets Yes Irregular path

Voidage

Mean value Non-uniformity Mean radial pro;le e–q graph

Low Small Zero Convex

High Large Non-zero Concave

Liquid ?ow

Scales excited Circulations Mean radial pro;le

∼ bubble size No Zero

∼ column size Yes Non-zero

Boundaries

Importance

High

Low

like in HoR, but the long-time radial pro;les of velocity and voidage are not ?at, unlike in HoR, and display roughly parabolic dependence on the column radius with a maximum on the centreline (Franz, Borner, Kantorek, & Buchholz, 1984). HoR is stable to small voidage and velocity disturbances and the transition to HeR with increasing gas ?ow is not a sudden event. The transition proceeds slowly and is indicated by an increasing number of coherent structures (circulations, vortices, i.e. ‘germs’ of HeR) of increasing size and intensity within the bubble bed. As correctly pointed out by Mudde and van den Akker (1999), in terms of the occurrence of coherent structures, the transition is quite a smooth process. However, the small-scale structures in HoR do not contribute to the resulting long-time average. Only if the structures reach a size comparable with the column size, they are oriented by the boundaries and produce coherence also on long-time scales, the well-known circulation loop. The transition is intermittent (both in space and in time) in nature and both HoR and HeR coexist in the column. A suitable measure of the column portion occupied by HeR would be an intermittency factor, which would re?ect the increase of the population of coherent structures with gas ?ow. The two basic regimes can simply be identi;ed from the character of the experimental voidage-gas ?ow rate plot, the e–q graph. The homogeneous voidage increases progressively (convex graph) with gas ?ow, while the heterogeneous voidage, no matter whether it is PHeR or THeR, follows a rational function (concave graph). The transition branch connects the convex

and concave graphs. It de?ects from the homogeneous branch at the in?ection point, passes through a maximum, and declines to the pure heterogeneous branch, see Fig. 1. The regime transition has been studied for a certain time. Experiments were performed, the e–q dependence was measured (Reith, Renken, & Israel, 1968; Ohki & Inoue, 1970; Yamashita & Inoue, 1975; Bach & Pilhofer, 1978; Maruyama, Yoshida, & Mizhushina, 1981; Zahradnik et al., 1997), and correlations for voidage and gas ?ow rate were published (Shah, Kelkar, Godbole, & Deckwer, 1982; Tsuchiya & Nakanishi, 1992; Wilkinson, Spek, & van Dierendonck, 1992; Jamialahmadi & Muller-Steinhagen, 1993; Reilly, Scott, De Bruijin, & MacIntyre, 1994; Sarra;, Jamialahmadi, Muller-Steinhagen, & Smith, 1999). The transition was identi;ed by analyzing pressure signals (Draho-s, Zahradnik, Puncochar, Fialova, & Bradka, 1991; Letzel, Schouten, Krishna, & van der Bleek, 1997; Lin, Tsuchiya, & Fan, 1999; Vial et al., 2000), liquid velocity signals (Lefebvre & Guy, 1999; Mudde & van den Akker, 1999), and by means of the drift-?ux concept (Lockett & Kirkpatrick, 1975; Shah et al., 1982; Kelkar et al., 1983; Zahradnik et al., 1997; Krishna, Ellenberger, & Maretto, 1999; Lin et al., 1999; Sarra; et al., 1999; Vial et al., 2000). Several models for the transition have also been suggested based on diAerent grounds: e.g. bubble drag force (Riquarts, 1979), gas phase slip velocity (Joshi & Lali, 1984), energy balance of the gas– liquid mixture (Gharat & Joshi, 1992b), or the concept of small and large bubbles (Krishna, Wilkinson, & van Dierendonck, 1991; Hyndman, Larachi, & Guy, 1997).

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However, all these models rely heavily on various empirical formulas and correlations, treat HoR and HeR separately, and ignore the transition region. The predictive value of these models is rather poor. To our knowledge, there is only one study on linear stability analysis of the homogeneous ?ow regime by Shnip, Kolhatkar, Swamy, and Joshi (1992), which yields a predictive criterion for the transition. Our approach is based on a simple and transparent physical model of the hydrodynamic coupling between the gas and liquid phases. The model re?ects the key feature of the e–q behaviour in bubble columns in steady state, and describes the regime transition as one smooth event. The model yields a criterion for the regime transition that agrees with both the drift-?ux model and the linear stability results. The model agrees well with experimental data from air–water bubble columns and represents a good platform for studying more complex systems.

2. Model The assumptions employed in the model are the following: (1) Voidage e, gas phase rise velocity (swarm velocity) u, and liquid velocity v in the column take steady and space–time averaged values. Since the mean liquid ?ux in the column is zero, u is also bubble slip velocity. (2) Bubble size and velocity w represent the averages over the bubble population and take constant (but diAerent) values for both regimes. (3) Gas phase rise in HoR is delayed by the liquid down-?ow. (4) Gas phase rise in HeR is enhanced by the liquid up-?ow. (5) Both HoR and HeR coexist in the column in a certain proportion in the transition regime. The basic formula for steady voidage in a column comes from the gas continuity equation. Equating the gas in- and out-?ows, Q and euS, the voidage is e = q=u;

(1)

where q is the speci;c gas ?ow, the external control parameter, and u is the actual gas phase rise velocity in the column. Note that e is de;ned as the ratio of two characteristic time scales 1=u and 1=q related to time of gas phase passage through the column and time of its supply. The task is to ;nd a suitable expression for u, which comprises the eAect of the induced liquid motion and bubble–bubble interactions mediated by the liquid. If the gas and liquid phases are uncoupled, u equals the

terminal velocity of an isolated bubble u = w;

(2)

and e increases linearly with q. Therefore, any deviation from linearity of the e–q plot is a measure of coupling between the phases. Our model expresses this coupling in a simple way as u = w ± correction:

(3)

2.1. Homogeneous regime 2.1.1. Formula for voidage In this regime, the correction in Eq. (3) is given by the liquid down-?ow. It is known that each rising bubble carries a certain amount of liquid, the drift volume (Darwin, 1953). This volume is expressed as a portion a of the bubble volume (Lighthill, 1956; Benjamin 1986). All bubbles thus generate the liquid up-?ow aQ. Owing to zero net liquid ?ux through the column, this liquid returns down with velocity v
through the cross-section area (1 − (1 + a)e
)S thus giving the down-?ow v
(1 − (1 + a)e
)S. Equating the liquid up- and down-?ows, the downward liquid velocity is aq : (4) v
= 1 − (1 + a)e
Following Assumption (3), the actual gas phase rise velocity is given as u
= w
− v
:

(5)

Putting (4) and (5) into (1), the homogeneous voidage √ q + w
− B
; e = 2(1 + a)w

  q 2 q
e ≈ +a + · · · for small e (6) w
w
then results from a quadratic equation with the discriminant


B = q2 + w 2 − 2(2a + 1)qw
:

(7)

For small e → 0, the linear term in expansion of Eq. (6) correctly recovers Eq. (2), and a stands for the correction term. Thus the leading-order term of the phase coupling in diluted suspensions is quadratic in q. 2.1.2. Critical values Setting the discriminant B to zero provides the critical gas ?ow predicted by the model, at which the transition occurs:  (8) q∗ = Zw
; Z = 1 + 2a − 2 a + a2 : The condition B 6 0 re?ects the unbalance between the upward and downward liquid ?ows: the latter becomes

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Fig. 2. Homogeneous regime: model prediction of voidage e(q). (a) EAect of terminal bubble velocity w (Eq. (6)). (b) EAect of bubble drift coe=cient a (Eq. (6)). Dotted line: zero coupling between gas and liquid phases (a = 0) (Eqs. (1) and (2)).

less than the former as the down-?ow area goes to zero with increasing voidage (?ooding). This is a purely kinematic criterion that predicts the highest possible value of q till which HoR may, in principle, exist, if absolutely stable. In real systems, the transition occurs at a lower value q1 ¡ q∗ , due to the dynamic instability of HoR. The maximum possible gas holdup in bubble columns predicted by the model is given by putting Eq. (8) into Eq. (6): e∗ =

1+Z : 2(1 + a)

(9)

Note that e∗ depends on a only. The homogeneous voidage given by Eq. (6) is plotted in Fig. 2 to show the eAect of the two parameters, w
and a. Gas holdup increases with gas ?ow roughly quadratically so that the deviation from linearity is positive (convex graph). A departure of measured data from this quadratic trend indicates the end of HoR and the beginning of the transition. Each line in Fig. 2 starts at q = 0 and ends at a limit point [q∗ ; e∗ ]. The parameter w
increases the extent of the homogeneous range, i.e. stabilizes HoR. The parameter a increases the deviation from linearity and substantially reduces both the extent of the homogeneous range and the maximum gas holdup. The straight line in Fig. 2b corresponds to zero coupling between gas and liquid phases. Fig. 3 shows the critical gas ?ow predicted by Eq. (8) and the corresponding maximum

Fig. 3. Homogeneous regime. Thin lines in (a) – (c): model prediction of critical gas ?ow q∗ and critical voidage e∗ . (a) EAect of terminal bubble velocity w (Eq. (8)). (b) – (c) EAect of bubble drift coe=cient a (Eqs. (8) and (9)). Bold lines in (a) – (c): results of linear stability analysis by Shnip et al. (1992) (Eq. (21) with a = 0:4 in (a), w = 0:2 in (b), and column geometry of case A in Table 2) (see Section 3.2 for details).

holdup predicted by Eq. (9). The apparent adverse eAect of bubble drift a on HoR is clearly seen. 2.1.3. Parameters The homogeneous voidage depends on two parameters, w
and a, whose values can be extracted from experimental data. The former represents a velocity scale for the motion of the gas phase in HoR and is obvious, with much data available in literature. The latter represents the strength of the coupling between the gas and liquid phases, and comprises the total information about the liquid velocity ;eld, boundaries, bubble arrangement and interactions. For an isolated particle in an unbounded

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and inviscid ?uid, the drift volume and the added mass coincide, and the drift coe=cient equals 1=2 (Darwin, 1953). Drift in a bounded inviscid ?uid, where the ?uid re?ux exists, has been studied by Eames, Belcher, and Hunt (1994), and near-wall drift by Eames, Hunt, and Belcher (1996). At ;nite Reynolds number, the inviscid eAects are weaker, but the liquid carried in the boundary layer and wake also contribute to a (Weber & Bhaga, 1982). Drift in many-particle systems has been studied by Kowe, Hunt, Hunt, Couet, and Bradbury (1988), and in multiphase ?ows by Tsuchiya, Song, Tang, and Fan (1992). There is no simple formula for a, and it must be determined from experimental data. Due to an intimate relation between drift and added mass, the values and trends of the latter, which is more elaborated, could possibly serve as a rough guideline for the former. For particles in general positions, the added mass ;rst increases with concentration (e.g. Zuber, 1964; Cai & Wallis, 1994; Spelt & Sangani, 1998). However, it must fall to zero as e → 1 for space reasons, e.g. foam. At large voidage, the ratio (lifted liquid=rising gas) ∼ (1=2)(1 − e)=e. The added mass strongly depends on bubble arrangement. For instance, it increases with concentration for side-by-side arrangement but decreases for vertical in-line arrangement (Hel;nstine & Dalton, 1974). The drift is, therefore, expected to depend on spatial con;guration of bubbles too. It means that by controlling the bubble arrangement in columns, we can, in principle, control the gas holdup. Note that the Darwinian drift generated by a particle moving through a ?uid considered here is not the well-known drift-?ux model developed by Wallis (1969). Wallis’s ‘drift’ refers to properly de;ned ?uxes of the phases relative to the two-phase mixture, and the concepts of particle added mass and particle drift volume are not employed. Actually, these two concepts should be equivalent since they both stem from kinematic considerations and are closed by suitable expressions for slip velocity. Our formula for slip velocity, Eq. (10) below, contains the drift coe=cient explicitly. 2.1.4. Formula for slip velocity The general eAect of bubble drift in a bounded region is generating upward and downward liquid ?ows, the latter delaying the bubble rise. A straightforward by-product of application of this powerful concept is a new non-empirical formula for bubble slip velocity. Putting Eq. (4) with q = eu into Eq. (5) the relation between u and e reads as ae : (u=w ≈ 1 − ae u=w = 1 − 1−e + · · · for small e; e0 = 1=(1 + a)):

(10)

The slip velocity decreases with bubble concentration and falls to zero at e0 , see Fig. 4a. The new formula (10)

presents an alternative to numerous empirical relations (e.g. Shah et al. 1982; Zahradnik et al., 1997; Sarra; et al., 1999). These are usually considered in the general form u=w=(e) with various expressions for the decreasing function (e). The most popular is that of Richardson and Zaki (1954) (RZ),  = (1 − e)m with empirical m ≈ 1:4. It is instructive to compare our formula (10) with the formula of RZ, see Figs. 4a and b. While the RZ graph is convex for the usual case m ¿ 1, our graph is concave. While the RZ formula gives unrealistic e0 = 1, our formula gives e0 6 1. While the RZ graph approaches e0 at zero derivatives of all orders, our graph has a non-zero tangent-(2 + a + 1=a). Comparing the RZ leading-order expansion for small e; u=w ≈ 1 − me, with our expansion, u=w ≈ 1 − ae, the purely empirical parameter m acquires a clear physical meaning. For diluted suspensions m is the drift coe=cient a. Therefore, a should be close to m at e → 0. Various values of m are reported in literature, e.g. m ≈ 1:4. Note that Zuber’s (1964) added mass coe=cient ≈ 0:5 + 1:5e has a comparable numerical coe=cient 1.5 at the leading-order term. The decrease of slip velocity with particle concentration re?ects the empirical fact that motion of particles in general positions is usually delayed by the hindrance eAect. Note, however, that the collective velocity of particles arranged in special positions can increase substantially with the concentration, e.g. due to the shielding eAect at in-line arrangement (Ruzicka, 2000). Putting e = q=u into Eq. (10), a useful relation between u and q results (see Figs. 4c and d):  u(q) = (1=2)(w + q + w2 + q2 − 2(1 + 2a)wq): (11) 2.2. Heterogeneous regime 2.2.1. Formula for voidage In HeR, the mean up-?ow with high bubble concentration occupies the large central part of the column, while the mean down-?ow with low bubble concentration occupies the small outer part. Therefore, the bubble transport by the down-?ow can be neglected and only the upward bubble advection can be considered. Following Assumption (4), the actual gas phase rise velocity is given by the terminal bubble velocity w

enhanced by the upward circulation velocity v

, u

= w

+ v

;

(12)

and we have to ;nd an expression for v

. Near the onset of the circulation, for q ≈ q1 , the unknown functional relation between v

and q can formally be expressed by a power series as v

= f(q) ≈ b + c(q − q1 ) + d(q − q1 )2 · · · :

(13)

At the ;rst approximation, only the linear part of the expansion is retained. Since the gas ?ow is the driving

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Fig. 4. Homogeneous regime: model predictions of u(e) and u(q). (a) New relation for slip velocity u(e) (Eq. (10)): eAect of bubble drift coe=cient a. (b) Old relation for slip velocity u(e) (Eq. u=w = (1 − e)m by Richardson & Zaki, 1954): eAect of empirical parameter m. (c) and (d) New relation for slip velocity u(q) (Eq. (11)). (c) EAect of terminal bubble velocity w. (d) EAect of bubble drift coe=cient a.

force of the liquid motion, the constant term must be zero and the leading-order term positive: v

= cq;

c ¿ 0:

(14)

Substituting Eqs. (12) and (14) into Eq. (1), the heterogeneous voidage becomes  q  q (e

≈ e

=

w + cq w

 q 2 −c + · · · for small e): (15) w

For small e → 0, the linear term in the expansion of Eq. (15) correctly recovers Eq. (2), and c stands for the correction term. For large q; e → 1, which would not happen if the quadratic term is retained in Eq. (13). Fig. 5 shows Eq. (15) for several values of parameters w

and c. Gas holdup follows the rational function dependence and the deviation from linearity is negative (concave graph). Expectedly, both parameters reduce the gas retention time in the columns, hence the holdup. 2.2.2. Parameters The heterogeneous voidage depends on two parameters, w

and c, whose values can be extracted from experimental data. The former represents a velocity scale for gas phase motion in HeR. The latter represents the

Fig. 5. Pure heterogeneous regime: model prediction of voidage e(q). (a) EAect of terminal bubble velocity w (Eq. (15)). (b) EAect of coupling constant c (Eq. (15)). Dotted line: zero coupling between gas and liquid phases (c = 0) (Eqs. (1) and (2)).

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strength of coupling between the phases. The numerical values of w

and c are the same for both THeR and PHeR and advantageously can be obtained from PHeR data, which are simple, monotonous, without transition. w

is usually larger than w
because bubble size increases with gas ?ow rate. This is re?ected in the concept of ‘small and large bubbles’ (e.g. Kolbel, Beinhauer, & Lengemann 1972; Vermeer & Krishna, 1981). Rational function expressions like Eq. (15) of various origins were suggested for voidage in HeR by other authors. Mashelkar (1970) used a purely empirical formula to describe a broad range of data collected from literature and found c=2. Reith and Beek (1968) derived a semi-empirical formula with c=2. The general theory for two-phase ?ows known as the drift-?ux model (Zuber & Findlay, 1965; Wallis 1969) has also been applied to HeR in bubble columns (Kelkar et al., 1983; Shah & Deckwer, 1983; Gharat & Joshi, 1992a,b; Deshpande, Dinkar, & Joshi, 1995; Zahradnik et al., 1997), yielding the same result, Eq. (15) with c close to 2. No physical meaning can be given to c in empirical models. In the drift-?ux model, c relates to radial distribution of velocity and voidage, and re?ects the shape of the averaged radial pro;les. Note that the long-term voidage non-uniformity is the driving force of liquid circulations. In our model, c = (v

=q) by Eq. (14), is the (eAect=cause) quotient, and thus relates to the driving force. 2.3. Transition regime 2.3.1. Formula for voidage The fact that both HoR and HeR coexist in the column during the transition naturally leads to the introduction of the intermittency factor p ∈ 0; 1 . The heterogeneous ?ow occupies p-portion of the column and the homogeneous ?ow (1 − p)-portion. Following Assumption (5), the gas velocity is expressed as a combination of expressions for HoR and HeR weighted by the factor p: u


= w
− (1 − p)v
+ pv

:

(16)

Substituting Eq. (16) into Eq. (1) gives the formula for the transition voidage: e


=

w


q : − (1 − p)v
+ pv



(17)

2.3.2. Parameters The new functional parameter p(q), the intermittency factor or the transition function, is the ;fth and last parameter of the model, whose values can be extracted from experimental data. p generally increases with q, departing from zero at the beginning q1 of the transition range q1 ; q2 , and reaching unity at its end q2 . Parameters of the same kind are commonly used to describe gradual

transitions between two diAerent modes of behaviour, e.g. laminar and turbulent (Tritton, 1988). The model basically consists of the above derived formulas for e(q) dependence. The prediction for HoR is given by Eq. (6), the prediction for PHeR by Eq. (15), and the prediction for the transition regime by Eq. (17). The prediction for the whole regime transition from HoR to THeR follows Eq. (6) for q ¡ q1 , Eq. (17) for q1 6 q 6 q2 , and Eq. (15) for q ¿ q2 . The model has ;ve parameters with clear physical meaning: (i) terminal bubble velocity w
in HoR, (ii) terminal bubble velocity w

in HeR, (iii) bubble drift coe=cient a, (iv) coupling constant c, and (v) intermittency factor p. At this stage of the model development, the parameters must be calculated from experimental data with the hope of future correlations. The range of the model validity (roughly q = 0:01– 0:30 m=s) is limited from below by the plate stable working regime, where all ori;ces on the plate are active (e.g. Zahradnik & Kastanek, 1979), and from above by plug (slug) regime (e.g. Whalley, 1987). Since the model departs from the mass conservation equation of gas phase, Eq. (1), and the equilibrium dynamics is introduced via the Darwinian drift concept, the stability condition q ¡ q∗ by Eq. (8) is a kinematic stability condition. 2.4. Calculation of model parameters from experimental data e(q) 2.4.1. Homogeneous regime Putting Eqs. (4) and (5) into Eq. (1) and linearizing gives


1 e







= (1 + a) +

1 w





 q(1 − e
) ; e
2

(18)

yielding (1 + a) as the intercept and 1=w
as the tangent. Besides this, w
can be measured directly from a video record or found from bubble size in literature. A typical bubble in HoR is 4 –5 mm large and rises at about 20 cm=s. Since a can vary with e, Eq. (18) may not always give good results and a must then be found by trial and error to ;t the data. 2.4.2. Heterogeneous regime Linearization of Eq. (15) gives (q=e

) = (w

) + (c)(q);

(19)

for w

and c. These two parameters can easily be determined from PHeR data and then applied to THeR data. Again, w

can be found independently by direct measurement or from published data. A typical bubble in HeR is 5 –10 mm large and rises at about 25 cm=s. Eq. (19) usually works well.

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2.4.3. Transition regime Knowing w
; a; w

, and c, the remaining parameter can be calculated directly by rearranging Eq. (17): p(q) =

q=e


+ v
− w
: v
+ v



(20)

Here, v
(e
) is given by Eq. (4), e
by Eq. (6), and v

by Eq. (14). Eq. (20) is applied to the whole data range, but only during the transition interval p lies between 0 and 1 thus indicating the transition. 2.5. Characteristic values of e(q) graph 2.5.1. Gas cation of the model The model was tested by experimental data from Ho– He regime transitions in air–water bubble columns in our laboratory. Eight experimental runs were carried out in cylindrical containers of various dimensions, cases A–D in Table 2. Four runs were carried out with plates of type I to obtain the whole transition from HoR to HeR, and four with plates of type II to obtain PHeR as the reference

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state. The metal plates were made of brass and were 3 mm thick. Type I had ori;ces of 0:5 mm with pitch 10 mm, and type II had ori;ces of 1:6 mm with pitch 30 mm. To obtain consistent results, the relative plate free area was identical for all plates, 0.2%. The gas ?ow was measured by rotameters, and the voidage was determined from the bed expansion and=or static pressure diAerence along the columns. In each experimental run, the gas ?ow was set, the voidage was recorded after a certain time (several minutes) to reach a steady state, and then the gas ?ow was increased. Each of the eight runs was repeated ;ve times and the voidage values were averaged (relative error less than 5%). The PHeR data from plates II gave values of w

and c. The Ho branch of the transition data from plates I gave values of w
and a. The rest of the transition data gave the transition function p(q). The values of four model parameters, w
; a, w

; c, extracted from the experimental data are shown in Table 2. The values of the ;fth parameter, the transition function, are displayed in Fig. 6. Model predictions are compared with the experimental data in Figs. 7–9. The model predictions are represented by lines, the experimental data by disconnected marks. Comparison for PHeR is shown in Fig. 7(a). Comparison for HoR (more precisely: Ho branch of the whole transition curve) is shown in Figs. 8(a) – (d). Comparison for the whole transition curve is shown in Figs. 9(a) – (d). Fig. 8 and 9 show the same experimental data. Characteristic values of the e–q graph (the beginning and end of the transition interval, the point of the actual maximum) are listed in Table 3. 3.2. Model comparison with results of other authors Our model displays the following key features, which will be compared with results of other authors already published in literature. HoR (i) original application of the concept of Darwinian drift (Eq. (4)), (ii) quadratic leading-order dependence e ∼ q2 (Eq. (6)), (iii) predictive kinematic criterion for critical gas ?ow q∗ and voidage e∗ (Eqs. (8) and (9)), (iv) new formula for slip velocity (Eq. (10)), HeR: (v) purely formal and simplest derivation of the standard formula (Eq. (15)), Transition: (vi) considered as a gradual process, (vii) original application of the concept of intermittency factor (Eq. (16)), (viii) formulas for e(q) for the whole transition curve (Eqs. (6), (15), (17)),

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Table 2 Experimental conditions for cases A–Da Experimental conditions

Model parameters HoR (Plate I)

PHeR (plate II)

Case

H (m)

D (m)

w (m=s)

a (—)

w (m=s)

c (—)

A B C D

0.25 0.5 1 2

0.15 0.15 0.15 0.29

0.20 0.20 0.22 0.22

0.25 0.30 0.35 0.35

0.261 0.252 0.247 0.257

2.20 2.53 2.64 2.03

a H —ungassed liquid height, D—column diameter. Values of model parameters extracted from measured data. HoR (plate type I): w  — terminal bubble velocity, a—bubble drift coe=cient. PHeR (plate type II): w —terminal bubble velocity, c—coupling constant. PHeR values apply also to THeR.

Fig. 6. Transition functions p(q) calculated from transition experimental data (Eq. (20)) (connected marks). The experimental e–q data shown by marks in Fig. 9, other parameters in Table 2.

(ix) identi;cation of three important points of e(q) graph (points e1; 2; m and q1; 2; m ). The following authors developed diAerent approaches to the regime transitions, and their models are brie?y discussed. 3.2.1. Riquarts (1979) For HoR, the concept of slip velocity of bubble swarm is applied. The end of the regime is identi;ed with the ?ooding point. For HeR, the concept of large bubbles responsible for destroying the homogeneity is used. 3.2.2. Joshi and Lali (1984) HoR is approached via bubble slip. The homogeneity is deteriorated by turbulence of liquid circulations. HoR breaks with increasing turbulence intensity, when the characteristic turbulence length scale reaches the column dimension.

Fig. 7. Pure heterogeneous regime: model comparison with experimental data (cases A–D in Table 2, plate type II). (a) Comparison: data (marks), model predictions (lines) (Eq. (15)). (b) Determination of parameters w and c (Eq. (19)).

3.2.3. Gharat and Joshi (1992b) On the basis of energy balance, pressure drop is evaluated for both HoR and HeR regimes. The transition occurs when the He pressure drop is lower than the Ho pressure drop. The above models diAer from ours in the assumptions employed, concepts applied, and parameters used. They consider HoR and HeR separately and yield no explicit stability criteria. It is, therefore, di=cult to compare them with our model.

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Fig. 8. Homogeneous regime: model comparison with experimental data (cases A–D in Table 2, plate type I). Data (marks). Model predictions of homogeneous branch (lines) (Eq. (6)).

Fig. 9. Transition from homogeneous to heterogeneous regime: model comparison with experimental data (cases A–D in Table 2, plate type I). Data (marks). Model predictions of whole transition curve (bold full lines) (Eqs. (6), (15), (17)). Dotted lines—particular model predictions: (upper) homogeneous regime (Eq. (6)), (lower) pure heterogeneous regime (Eq. (15)), (middle) zero coupling between gas and liquid phases (a = c = 0) (Eqs. (1) and (2)).

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Table 3 Characteristic values of gas ?ow q and voidage e during regime transitiona Characteristic values of gas ?ow and voidage (plate I) Gas ?ow (m=s)

Voidage [—]

Case

q∗

q1; 1

q1; 2

q2; 1

qm

e∗

em

e1; 2

A B C D

0.076 0.070 0.072 0.072

0.065 0.049 0.045 0.042

0.062 0.050 0.045 0.045

¿ 0:15 ¿ 0:16 0.129 0.103

0.086 0.079 0.073 0.075

0.55 0.52 0.49 0.49

0.42 0.34 0.31 0.33

0.38 0.28 0.23 0.21

a Gas ?ow: q∗ —theoretical critical value (model), q 1; 1 and q1; 2 —actual critical values (two diAerent estimates of the beginning of the transition range; experiment), q2; 1 —actual value of the end of the transition range (experiment); qm —gas ?ow at maximum voidage. Voidage: e∗ —theoretical maximum value (model), em —actual maximum value (experiment), e1; 2 —voidage at q1; 2 (experiment).

Fig. 10. Comparison with Krishna’s model (Krishna et al., 1991): experimental data (marks), our model (thin line), Krishna’s model (bold line). Parameters: ours (Table 2), Krishna’s (B = 1 and C = 0:8 from the original paper; qcrit = 0:042 from our model, Table 3).

3.2.4. Krishna et al: (1991) Their model consists in approximating HoR with a straight line and HeR with a curved line: eHo = 4q; q ¡ qcrit ; eHe = eHo crit + B(q − qcrit )C ;

q ¿ qcrit :

Both relations are rather coarse approximations and the transition regime is disregarded completely. The empirical parameters B and C lack any physical meaning. The critical gas ?ow qcrit is not a result but an input of the model. The model is based on the concept of small and large bubbles originated earlier (Kolbel et al., 1972), where the extra He voidage is ascribed to the presence of large bubbles (see also Shah & Deckwer, 1983; Deckwer, 1992). This concept ;nds its support in disengagement experiments (e.g. Hyndman et al., 1997). Their model is compared with our model in Fig. 10. Their model is too coarse to comprehend the ;ne structure of the regime transition for which our model has been developed. 3.2.5. Drift-
velocities and concentrations of the phases. It looks at relative motion of phases and is suggested for ?ows with ?at radial pro;les. The basic quantity is the so-called drift ?ux j ≡ ue(1 − e), or j = q(1 − e) by Eq. (1), which is the gas ?ux through a surface moving with the speed of the mixture. The dynamic input into this essentially kinematic concept is the slip velocity, u(e), which also presents the closure of the model. The model is used as follows. With an empirical expression for the slip speed, the theoretical curve jT (e)=ue(1 − e) is drawn. Then, the experimental values jE (e) = q(1 − e) are calculated from measured q and e and are plotted in the same graph. In HoR, jE = jT , and the transition occurs where this equality breaks. This concept is commonly used for indicating the regime transition in bubble columns (Wallis, 1969; Lockett & Kirkpatrick, 1975; Shah et al., 1982; Kelkar et al., 1983; Shah & Deckwer, 1983; Krishna, Ellenberger, & Maretto, 1999; Lin et al., 1999; Sarra; et al., 1999; Vial et al., 2000). To show the compatibility of our results with Wallis’s concept, the drift-?ux plots for our data (marks) are displayed in Figs. 11(a) – (d). The theoretical curves (lines) were calculated using our new expression for the slip speed, Eq. (10). The data follow the lines well and depart at critical values that coincide with those obtained from our model, see values of e1 in Table 3. Note that for bubble columns with zero net liquid ?ux, the original Wallis’s critical point is the theoretical point where the line ;xed in the right lower corner touches the parabola as shown in Fig. 11(a). 3.2.6. Drift-
M. C. Ruzicka et al. / Chemical Engineering Science 56 (2001) 4609–4626

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Fig. 11. Comparison with the drift-?ux model for homogeneous regime (Wallis, 1969): theoretical drift-?ux plot (parabolic lines), experimental drift-?ux plot (marks). Departure of marks from lines indicates transition. Two arrows in (a) represent two other criteria for the transition (see Section 3.2). Results in Table 5.

Fig. 12. Comparison with the drift-?ux model for heterogeneous regime (Zuber & Findlay, 1965; Wallis 1969). Theoretical line: Zuber–Findlay ;t and our Eq. (19). Full marks: experimental data from pure heterogeneous regime. Empty marks: experimental data from regime transition. Results in Table 5.

(Zahradnik et al., 1997; Vial et al., 2000): the HeR data lie on the ;t by Eq. (19) but the HoR and transition data depart. The point where the transition data join the He ;t represents the end of the transition range. Our results are compatible also with this approach. Fig. 12 shows the Zuber–Findlay ;t (line), the PHeR data following the ;t (full marks), and the transition data (empty marks) joining the line at the end of the transition at about q=0:1. This value agrees well with the value q2; 1 =0:103 obtained from the transition functions of our model, (see Table 3).

3.2.7. Shnip et al: (1992) The linear stability analysis is performed with a set of simple two-phase equations of motion for bubbly mixture (both mass and momentum) in a two-dimensional bubble column and a dynamic stability criterion for HoR is obtained:   sinh( A) 2g ¡ : (21) YPJ
w D cosh( A) − 1 Here, g is the acceleration due to gravity and A the column aspect ratio H=D. YP relates to the pressure drop across the plate and equals KV + 2KT q=w, where parameters KV and KT link to the relative plate free area ’ via empirical expressions, KV = 0:98’−1:5 =%w; KT = 0:30’−2 =%; % being the liquid density. J
= u + e(@u=@e) is the concentration derivative @J=@e of the ?ux J = eu. Note that J
equals the propagation speed of concentration kinematic waves, and that J
= 0 is the kinematic instability condition (e.g. Zuber, 1964). Actually, J
is the factor that, going to zero, increases the l.h.s. of Eq. (21) thus breaking the stability. Criterion Eq. (21) predicts the following: (I) increase of qc with increasing w, (II) decrease of qc and ec with increasing a, (III) decrease of qc and ec with increasing H . The predictions (I) and (II) (bold lines) are compared with our results (thin lines) in Figs. 3(a) – (c). The slip

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Table 4 Comparison of critical values of gas ?ow qc and voidage ec at the transitiona Comparison of predicted and experimental critical values of qc and ec Critical gas ?ow qc (m=s)

Critical voidage ec [-]

Case

Model

Shnip ’92

Experiment

Model

Shnip ’92

Experiment

A B C D

0.076 0.070 0.072 0.072

0.075 0.069 0.070 0.067

0.065 0.049 0.045 0.042

0.55 0.52 0.49 0.49

0.51 0.47 0.44 0.39

0.38 0.28 0.23 0.21

a (i) Model: values of q∗ and e∗ predicted by Eqs. (8) and (9) of our model. (ii) Shnip et al. (1992): values predicted by stability criterion Eq. (21). (iii) Experiment: our experimental values of q1; 1 and e1; 2 taken from Table 3.

velocity formula u(e) needed in Eq. (21) was taken from Eq. (10). Our model agrees with the theoretical stability results not only qualitatively, but to a surprisingly high degree also quantitatively, always giving slightly higher numerical values, see Table 4. In the rows of the table, the critical values order as follows: model ¿ Shnip’92 ¿ experiment. This ordering is physically signi;cant as it re?ects three diAerent kinds of stability of HoR considered: (i) absolute stability in our model, (ii) linear stability with respect to in;nitesimal disturbances in Shnip’s analysis, (iii) fully non-linear case with ;nite-amplitude disturbances in experiments. Obviously, the kinematic limits given by our model cannot be exceeded, so our criticals are the highest. Dynamic and non-linear eAects naturally restrict the stability range. In the columns of Table 4, a general ordering trend can be seen with respect to column height: case A ¿ case B ¿ case C. This is also signi;cant, as it re?ects the destabilizing eAect of the column height as predicted by Eq. (21), item (III). It can be concluded that our model and data agree well with the stability theory of Shnip et al. (1992). 3.2.8. Other criteria for regime transition The regime transition is also identi;ed by a sudden steep increase of the graph of experimental plot of u against q (e.g. Shah & Deckwer, 1983; Krishna et al., 1991; Hyndman et al., 1997). Our results plotted in this way are shown in Fig. 13. Because u was not measured directly in the experiments, Fig. 13 shows lines predicted by our model. The ordering of the lines complies with the decreasing trend of qc from cases A–D in Table 4. Sarra; et al. (1999) suggest a criterion related to the drift-?ux model, where the transition occurs at the minimum of (u − j)=u, which upon substitution for j equals 1 − e + e2 , plotted against q. Our results plotted in this way are shown in Fig. 14. The model predictions (lines) ;t the experimental points (marks) well. The minima occur somewhere in the middle of the transition range, and the ordering of the corresponding critical values of q for cases A–D is again in accord with Table 4. Lapidus and Elgin (1957) expect the transition when @J=@e = 0 and the critical point thus is the maximum of

Fig. 13. Comparison with criterion ‘Steep increase in u(q)’. Lines: model prediction of the dependence calculated as u=(1 − p)u +pu . Results in Table 5.

Fig. 14. Comparison with criterion ‘Minimum of (u − j)=u’ (Sarra; et al., 1999). Model prediction (lines) and experimental data (marks). Results in Table 5.

the graph of J (e). In our case, the points are apexes of the parabolas in Fig. 11, see the arrow in Fig. 11(a). Zuber (1964) shows that HoR breaks when kinematic concentration waves cannot propagate through the medium. Knowing that the speed of these waves equals @J=@e, it links to Lapidus’s result and opens the door to far-reaching area of waves in dispersed media.

M. C. Ruzicka et al. / Chemical Engineering Science 56 (2001) 4609–4626

3.2.9. Quadratic leading order dependence e ∼ q2 It is commonly believed that in HoR the dependence is e ∼ qr , where r varies from 0.7 to 1.2. This information appears in Shah et al. (1982) and is repeated by other authors (e.g. Reilly et al., 1994; Jamialahmadi & Muller-Steinhagen, 1993; Sarra; et al., 1999). It results in the general impression that the Ho voidage graph is not much convex and is close to a straight line (at least for air–water system). This is, however, not completely true. Firm visual evidence that r ¿ 1 comes from many experimental data scattered in literature. Theoretical considerations predict the convexity too (Wallis, 1969; lines 1– 4 in Fig. 14 of Sarra; et al., 1999). Our model predicts a quadratic leading-order dependence at low e by Eq. (6) and even a super;cial visual inspection of Fig. 8 indicates the convexity. As an example, the ;rst ;ve data points of the Ho branch in Fig. 8(a) were well ;tted with a parabolic trendline, e = 55q2 + 3:2q (rxy = 0:999). Generally, the curvature can be subtle, and ;ne and precise data are needed to resolve it. To illustrate the above discussed methods, concepts, and models, Table 5 summarizes numerical values of the criticals obtained from these diAerent sources. The comparison is made for our experimental case A (Table 2). The ;rst part of Table 5 (nos. 1–5) contains values determined from the experimental data by diAerent evaluating procedures. The second part (nos. 6 –9) presents purely theoretical predictions calculated using the parameter values of case A (Table 2). We are convinced that our model, no. 1 in Table 5, gives the correct values of the critical point where HoR loses stability and the transition starts. 4. Discussion 4.1. Model and data The very good agreement of the model predictions with the experimental data documented in Figs. 7–9 provides a strong substantiation of the model and its assumptions. In particular, it justi;es adopting the bubble drift concept for HoR, retaining only the linear leading-order term in the expansion in Eq. (13) for HeR, and expressing the intermittent character of the transition in terms of the transition function, Eq. (16). In Fig. 7, the cases A–D are close to one another, so HeR in our columns was not sensitive to the column dimensions, in contrast to HoR. It is reported that above 15 cm diameter, the column width does not aAect HeR much (Zahradnik et al., 1997). In Fig. 8, the model prediction de?ects from the data at the beginning of the transition, point [q1 ; e1 ], and ends in the theoretical critical point [q∗ ; e∗ ]. The distance between these two points is a measure of the importance of dynamic effects. These eAects are not considered in our model and reduce the stability range. In Fig. 9, the opposite eAects of the phase coupling in HoR and HeR are clearly seen

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from the positive and negative deviations of the voidage graph from the straight line. Cases A and B display more stable HoR, because the transition has not ;nished within the range of gas ?ow considered. Table 3 shows important values of gas ?ow and voidage that characterize the transition. Both estimates of the critical gas ?ow, q1; 1 and q1; 2 , (hence e1; 1 and e1; 2 ) are close to each other and reach 60 –80% of the theoretical limit q∗ . The point qm where the experimental voidage reaches the maximum is close to q∗ . The maximum experimental voidage em reaches 60 –75% of the limit value e∗ and is closer to it in lower columns. The sensitivity of HoR to the column height is apparent even from this limited set of data: the transition begins sooner and the maximum voidage is lower in higher columns, cases A–C. The eAect of the column width cannot be inferred from the present data, because only two values of the column diameter were applied. The present model does not involve the column dimensions H and D. It is because the model stems from the continuity equation that after scaling does not give any non-dimensional numbers, which could contain length scales. Also, no quantities related to length were introduced into the model ad hoc. The dimensions will enter the model via future correlations for its parameters. On the other hand, the dynamic Shnip’s theory contains the column dimensions as it is derived from the momentum equations that after scaling give dimensionless numbers containing H and D. Recently, an alternative stability theory of dispersed layers has been put forward (Ruzicka & Thomas, 2001) that also involves the dynamic aspects and the eAect of layer dimensions. It seems that we lack reliable and conclusive experimental data on the eAect of H and D on HoR stability. For the same reason, our model at its present stage does not involve physico-chemical characteristics of the phases. 4.2. Comparison with others authors Fig. 10 clearly shows that the model of Krishna et al. (1991) is too rough to re?ect the subtlety of the regime transition, and the comparison goes in favour of our model. Their model is very simple and is therefore useful for practical purposes. The drift-?ux model (DF) by Wallis is a well-established and reputable theory that gives reliable results. Our model agrees well with the homogeneous version of DF, see Fig. 11 and compare nos. 1 and 2 in Table 5, where the numerical values coincide. Moreover, the theoretical predictions of both models are identical, compare nos. 6 and 7 in Table 5. This ;nding supports the expected hidden equivalence of these two concepts, as mentioned earlier. In addition, our model agrees well also with the heterogeneous version of DF, see Fig. 12. The values for the end of the transition range obtained from the Zuber–Findlay plot and from our

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Table 5 Summary of predictions of the transition point from diAerent sourcesa Extracted from experimental data

Theoretical predictions

Source no.

1

2

3

4

5

6

7

8

9

qc (m=s) ec [-]

0.065 0.38

— 0.38

0.115 —

— 0.63

0.085 —

0.076 0.55

— 0.55

— 0.51

— 0.4

a Symbols:

qc —critical gas ?ow rate, ec —critical voidage. Evaluation of qc and ec from experimental data presented in this study by diAerent procedures: (1) our model (present study); (2) the drift-?ux model for homogeneous regime (Wallis, 1969); (3) the drift-?ux model for heterogeneous regime (Zuber & Findlay, 1965; Wallis, 1969); (4) ‘Steep increase in u(q)’; (5) ‘Minimum of (u − j)=u’ (Sarra; et al., 1999). Purely theoretical predictions: (6) our model (present study); (7) the drift-?ux model for homogeneous regime (Wallis, 1969), see the line and arrow in Fig. 11(a); (8) linear stability theory of Shnip et al. (1992); (8) zero derivative of ?ux (Lapidus & Elgin, 1957; Zuber, 1964).

transition function are the same, see Section 3.2. The concept of kinematic waves is closely related to DF (e.g. Wallis, 1969; Whitham, 1974) and gives a good estimate of the transition point, compare nos. 1 and 9 in Table 5. Our model agrees well with the theoretical study by Shnip et al. (1992). The ordering of the numerical values in rows and columns of Table 4 is highly physically relevant, see Section 3.2. Although Shnip et al. use very simple two-phase equations of motion, their predictions presented here compare favourably with our experimental data. It is likely that our data present the ;rst experimental test of Shnip’s theory. A useful up-to-date discussion of the equations of motion for bubble columns has recently been provided by Minev, Lange, & Nandakumar (1999). Simple and not very substantiated criteria like those in Figs. 13 and 14 are handy and give plausible results. They are, however, not suitable when the instability point must be determined precisely, compare nos. 4 and 5 with no. 1 in Table 5. The fact that our model agrees not only with the experimental data but also with both the drift-?ux model and Shnip’s analysis gives strong support to our work. Moreover, our concept oAers an added value in improving the current models and bringing new and original insights. Of the nine key features of our model listed in Section 3.2, all but item (iii) seem to be new. As for item (iii), the other transition criteria are presented in Table 5. Lastly, since Occam’s Razor principle was thoroughly applied during the model construction, the model has a very favourable ratio of (e=ciency=complexity). 5. Conclusions The minimum set of assumptions underlying the model introduced seems to be su=cient to generate a modelling concept that is able to comprehend the key features of voidage-gas ?ow behaviour in bubble columns operated under steady conditions. The model: • Gives formulas for description of the e–q dependence. • Explains deviations from the linear e–q dependence in

terms of phase coupling.

• Predicts the transition point and the maximum possible

voidage in bubble columns.

• Extracts the beginning and end of the transition range

from experimental data.

• Agrees with experimental data. • Agrees with results of other authors.

The ;ve model parameters have clear physical meaning, can be calculated from experimental data, and can be correlated with geometrical and physico-chemical properties of the system. Although very simple, the model has potential for further development along both (i) fundamental lines (e.g. elaborating drift and added mass concepts for moderate Reynolds number multiphase ?ows and the concept of spatio-temporal intermittency for complex ?ow systems) and (ii) practice lines (e.g. working out correlations for the parameters to establish useful and reliable criteria for design of gas–liquid contactors).

Notation a A c D e H j J p q Q S u

v w

bubble drift coe=cient: (drift volume)=(bubble volume), dimensionless column aspect ratio H=D, dimensionless coupling constant, Eq. (14), dimensionless column diameter, m voidage (gas holdup), dimensionless height of ungased liquid column, m drift-?ux, m=s ?ux (general), m3 =m2 s intermittency factor (transition function p(q)), dimensionless speci;c gas ?ow Q=S (super;cial gas velocity), m=s gas ?ow, m3 =s column cross-section area, m2 actual gas phase rise velocity (also: swarm, interstitial), m=s (for zero net liquid ?ux u is also mean bubble slip velocity) liquid phase velocity in the column, m=s, bubble terminal velocity, m=s

M. C. Ruzicka et al. / Chemical Engineering Science 56 (2001) 4609–4626

Indices 1 2 c i; j m ∗





; ;

beginning of the transition end of the transition critical value, transition point (general) diAerent estimates of qi; j and ei; j maximum value critical values (model) homogeneous, heterogeneous, and transition regimes

Abbreviations HoR HeR PHeR THeR

homogeneous regime heterogeneous regime pure heterogeneous regime heterogeneous regime resulting from transition

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