1 1.2 1.4 1.6 1.8 2 2.2 2.4
1 10 100 1000 10000 100000
C0
ReL
Laminar Turbulent
Figure 3.4: Prediction of C0 in terms of ReL according to the model of Collins et al. (Eq.3.26).
evolve as Ca2/3[13] so that C0tends to unity as confirmed by recent experiments [4,88].
3.2 Flow pattern maps
The distinguishing feature of two-phase flows in channels is the gas-liquid in-terface, which depends on some physical properties (density, viscosity, surface tension), flow rates of both fluids, gravity level and the hydraulic diameter of the channel. Even if there are many different interface topologies, two-phase flows can be classified into a limited number of flow patterns. Baker [5] plotted the first flow pattern maps in order to provide a satisfactory classification of flow patterns. Different parameters such as flow rates or superficial velocities were used as coordinates.
New questions were later addressed regarding the flow pattern boundaries, thus triggering interest in the study of flow pattern transitions. Govier and Aziz [36] summarized the early steps carried out in the 1960s, reporting the flow terns found both in vertical and horizontal tubes as well as results in flow pat-tern transitions. The concepts of stratified, bubble, slug and annular flows first appeared during this decade.
26 3 State of the art
Interested in the role played by gravity in two-phase pattern formation, Suo and Griffith [83] performed experimental work on air-water flow in minichan-nels (0.5 up to 0.7 mm internal diameter). They developed a criterion based upon the Eötvös number that determines when gravitational effects become negligible with respect to capillary effects:
Eo= ∆ρgφ
2
σ = buoyancy f orce
sur f ace f orce <0.29 (3.27) According to the authors, when this criterion is accomplished, gravitational effects become masked by superficial effects, resulting in the irrelevance of the channel orientation. Under these conditions, the same flow patterns and flow pattern transitions are expected in vertical and horizontal tubes, and two-phase flows can be assumed to be under microgravity-related conditions. In these conditions, stratified flows are not expected because they strongly depend on gravity forces.
More recently, Damianides and Westwater [21] reported experimental data with air-water flow in minitubes (1 up to 5 mm i.d.) and stated that gravitational effects become irrelevant for i.d. between 1 and 2 mm. Experimental data on two-phase-flows in microgravity conditions were first reported by the pioneer-ing work of Heppner et al. [38] on board the KC-135 zero gravity aircraft. Data under microgravity became more common in the late 1980s as a consequence of an increase in accessibility for researchers to drop towers and parabolic flights.
Most of the authors agree with the existence of three main flow patterns in microgravity, namely, bubble, slug and annular. Nevertheless, there is still dis-agreement concerning flow pattern transitions. Eastman et al. [25], Karri and Mathur [43] and Crowley et al. [20] attempted unsuccessfully to extend transi-tional models and flow pattern maps existing in 1g to the microgravity environ-ment. Lee [47], Zhao and Rezkallah [92], Reinarts [75] and Zaho and Hu [91]
approached the subject trying to explain the forces acting on the fluids during the transitions. Zhao and Hu [91] succeeded in describing the slug-annular tran-sition by developing flow pattern maps based on the Weber number. Dukler et al. [22], Colin, Fabré and Dukler [17], Colin, Fabré and McQuillen [18] and Bous-man [10] developed the so called semi-empirical void fraction model, which ex-plains the flow pattern transitions as a consequence of the coalescence between bubbles. McQuillen, Colin and Fabré [58] and Zhao [91] provided a remarkable summary of the previous studies on flow patterns, flow pattern transitions and other relevant topics concerning two-phase flows.
3.2 Flow pattern maps 27
With respect toward the bubble-slug transition, the semi-empirical void frac-tion model is the most widely used. By using the drift-flux model and assem-bling experimental data collected in microgravity (Colin et al. [17], Reinarts [74], Bousman [10]) and Colin et al. [18] developed a model capable of predicting the bubble-slug transition based upon the Suratman number. Recently, Sen [80] has established that Su is the key dimensionless group in the bubble-slug transition in microgravity, according to the analysis of the forces.
3.2.1 Suratman number based model
According to Colin et al. [17], bubble velocity can be reasonably well predicted in microgravity by a drift-flux model and expressed by:
UG =C0(USL+USG) (3.28)
In their experiments, the void fraction distribution coefficient ranges between 1.15 and 1.3. Hereafter, we will assume a C0 = 1.2. Dividing by USG, Eq.3.28 may be expressed as:
USL =USG1−C0α
C0α (3.29)
The authors consider the existence of a critical or transitional void fraction value, αC, that corresponds to the required void fraction at which the bubble-slug transition occurs. Thus, the bubble-bubble-slug transition may be expressed in terms of a critical value of the void fraction.
Dukler et al. [22] registered a set of experimental data in microgravity in or-der to clarify the mechanisms that give rise to flow pattern transitions. They predict the bubble-slug transition based on a bubble diameter criterion, assum-ing the transition occurs when the bubble diameter reaches the value of the tube diameter. The transition is then explained as a consequence of the coalescence phenomena. Coalescence events are related to the number of collisions between bubbles. According to the authors, these events depend on the bubble packing factor, which is not allowed to exceed 0.53. For the bubble-slug transition, an αC=0.45 is suggested, and Eq.3.29becomes:
USL =1.22 USG (3.30)
In contrast, experiments reported by Colin et al. [17], Bousman [10] with different i.d., and by Reinarts [74] with other fluids than air and water (namely R12) showed that the bubble-slug transition took place for a critical void fraction around 0.2.
28 3 State of the art
Taking into account the previous experimental data, Colin et al. [18] propose an approach to predict the bubble-slug transition. According to this approach, the transitional void fraction is related to the Suratman number. Two differ-ent regimes were pointed out for the coalescence mechanism, corresponding to two values of the critical void fraction. For low values of Su, the bubble-slug transition is controlled by bubble packing, which is known as the inhibiting co-alescence regime. At large Su values, the coco-alescence phenomenon is promoted due to effective collisions between bubbles along their flow path. The authors conclude that:
Su<1.5 · 106→αC=0.45 (3.31) Su>1.7 · 106→αC=0.2
This model reconciles both values of the critical void fraction reported in previous experimental work. Nevertheless, as stated by McQuillen et al. [58], it does not give any insight into the mechanism of the coalescence or the influence of the specific process used to generate the bubbles.