Vapor-Liquid Interfaces, Bubbles and Droplets: Fundamentals and Applications (Heat and Mass Transfer)
In great part of the saturated region, however, there is a superposition of equilibrium and real qualities, enabling the convenient determination of vapour flow rates from energy balances. A more detailed exploration of the definitions of quality was provided by Baehr and Stephan The classical i.
In Figure 3 , the heat transfer regimes in convective boiling are represented qualitatively as a function of enthalpy hence equilibrium quality and wall heat flux. Figure 4 presents the qualitative behaviour of the heat transfer coefficient as a function of equilibrium quality and of heat flux. The total mass flow rate and the pipe geometry length and diameter are assumed constant and, for simplicity, the pressure drop is assumed negligible. In Figures 3 and 4 , lines i to vii represent increasing heat flux conditions.
Curve i relates to a low heat flux condition, and together with curve ii , envelope the design operating conditions of two-phase heat transfer equipment e. Flow patterns within this range of heat fluxes are typically those observed in adiabatic gas-liquid flows i.
All of the non-equilibrium effects reported in this paper occur within the low heat flux range of the boiling region illustrated in Figures 3 and 4 curves i and ii. In this region, starting with heat transfer to saturated liquid near the entrance of the channel, subcooled boiling is initiated giving rise to an increasing heat transfer coefficient defined in this region as the rate of the wall heat flux to the difference between the wall temperature and the bulk temperature. After equilibrium saturation has been attained, the heat transfer coefficient now defined as the ratio of the wall heat flux to the difference between the wall temperature and the saturation temperature remains approximately constant and independent of quality reflecting the dominance of nucleate boiling effects.
Further downstream, as quality increases, there is a transition to a heat transfer mode dominated by forced convection and the heat transfer coefficient increases with increasing quality up to the point of critical heat flux CHF , characterized by the 'dryout' of the liquid film. Departure from the Classical Behaviour.
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As pointed out by Hewitt , the representation of the heat transfer coefficient behaviour delineated in Section 2 has been the basis of design calculations for forced convective boiling for many years, with correlations being developed to represent the various regions of Figures 3 and 4. In this section, the non-equilibrium phenomena responsible for substantial departures from this classical representation will be identified. Physical interpretations of these phenomena will be provided together with phenomenological models for their prediction.
The first case of departure from the classical behaviour of convective boiling heat transfer is the occurrence of heat transfer coefficient peaks in the region of near-zero equilibrium quality. The peaks were observed in boiling of hydrocarbons Kandlbinder, ; Urso et al. An example of the phenomenon of near-zero quality heat transfer peaks is depicted in Figures 5 and 6. The results were obtained by Kandlbinder , who was the first to investigate systematically this effect.
The experiments were performed in a 0. More recently, Urso et al. They aimed at obtaining a wider equilibrium quality range inside which sub-annular flow patterns bubble, slug and churn would persist over larger distances along the channel. Zones of heat transfer enhancement in the near-zero quality region were observed repeatedly. An explanation for the existence of near-zero quality peaks was pursued quantitatively by Barbosa and Hewitt According to the theory, in a situation where the conditions for bubble nucleation at the wall are poor, the layer of fluid adjacent to the wall becomes highly superheated.
Therefore, once a bubble is nucleated it grows rapidly, suddenly releasing the thermal energy stored in the surrounding liquid.
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Under some circumstances, the rate of change in void fraction associated with bubble growth may be high enough to trigger an abrupt flow pattern transition leading to the formation of a vapour plug Figure 7. The postulated mechanism for the formation of the vapour plug in the subcooled region is supported by experimental evidence by Jeglic and Grace , who studied the onset of flow oscillations in forced convective boiling of subcooled water at sub-atmospheric pressures in electrically heated tubes constant wall heat flux.
They observed that the flow oscillations were accompanied by a high rate of change in void fraction, whose association with the formation of a vapour slug was confirmed through visual observation of the flow structure. A detailed discussion concerning the mechanisms that favour the occurrence of near-zero quality peaks is given elsewhere Barbosa and Hewitt, In summary, the four mechanisms are i large vapour formation for a given superheat, ii low liquid thermal conductivity leading to large differences between the wall temperature and the local saturation temperature, iii high subcooling and iv low mass transfer resistance to bubble growth.
These mechanisms were found to be prevalent in the situations where the phenomenon of heat transfer peaks was identified in the literature. Section 4 will review the mechanistic model for non-equilibrium slug flow that predicts the heat transfer peaks in the near-zero quality region. The second case of departure from the classical behaviour of convective boiling heat transfer is the deterioration of heat transfer coefficient with increasing quality at high qualities typical of annular flow.
The heat transfer coefficient behaviour as a function of quality is shown in Figure 8 Kandlbinder, Similar behaviour was observed by many investigators for a number of binary and multicomponent mixtures in both vertical Celata et al. An extensive literature review was carried out by Barbosa In binary and multicomponent evaporating systems, the difference in volatility between the components gives rise to axial gradients of concentration and hence of saturation temperature in both liquid and vapour streams due to the preferential evaporation of the more volatile component s even when local component equilibrium occurs.
Figure 9 , adapted from Thome and Shock , illustrates the axial distributions of saturation and wall temperature for a mixture and for a single component undergoing phase change in a tube. As quality increases, the liquid phase becomes richer in the less volatile component and the fluid saturation temperature increases in many cases, overcoming the decrease associated with the negative axial pressure gradient. In the annular flow regime, without bubble nucleation at the wall, the heat transfer coefficient decrease associated with mixture effects takes place in different ways depending on the form of heating imposed to the surface Wadekar, In the present context, mixture effects are defined as the build-up of concentration gradients adjacent to the vapour-liquid interface resulting from the preferential evaporation of the more volatile component s.
This decreases the interfacial concentration of the lighter component, , to a value lower than the bulk liquid phase concentration,. For simplicity, ideal cases in which no droplet interchange entrainment exists are depicted in Figures These figures exhibit temperature profiles across the liquid film for prescribed wall temperature and heat flux boundary conditions, respectively.
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In both cases, profiles with and without mixture effects are shown. In the wall temperature controlled case Figure In the heat flux controlled case, the rise in the interface temperature does not change the temperature driving force. Rather, this potential must remain unchanged so that the product of the local film heat transfer coefficient and the temperature difference is equal to the applied heat flux 1.
The increase in wall temperature from to is illustrated in Figure So what causes the reduction of the heat transfer coefficient observed so systematically in the literature for the constant heat flux case? The answer is practicality. For engineering design calculations, it is more convenient to define in terms of than , where is generally determined experimentally.
Because is lower than , the ratio of the wall heat flux to the design based wall superheat is also lower. It is therefore expected that conventional prediction methods that do not take these effects into account will overestimate the experimental heat transfer coefficient. As will be shown in Section 5, in order to properly quantify and in the annular flow regime and consequently predict the heat transfer coefficient under forced convective boiling, two mechanisms must be accounted for, namely,.
Mass transfer resistance as a result of component s preferential evaporation and formation of concentration gradients adjacent to phase interface s in both phases;. A hydrodynamic non-equilibrium resulting from the difference in average concentration between the liquid film and the liquid entrained as droplets in the gas core. Item b above is a direct consequence of the processes of droplet entrainment and deposition; vigorous mass exchange phenomena which disrupt the hydrodynamic equilibrium of annular flow. Amongst other effects, these phenomena are known to exert a large influence on important flow parameters like pressure drop Hewitt and Hall-Taylor, Nevertheless, the significance of droplet interchange on forced convective boiling of mixtures seems to have been overlooked in previous studies and the correct prediction of this mechanism may hold the key to understanding the deterioration associated with the heat transfer coefficient at high qualities Barbosa and Hewitt, a, b; Barbosa et al.
Figure 11 depicts an interpretation of the physics of the mixture vaporization problem. In design calculations, the two-phase flow pattern is usually ignored and it is implicitly assumed that the whole of the liquid flow is available for evaporation Figure Usually, a flash calculation is used to determine the amount of liquid which has evaporated for a given wall heat flux and the saturation bubble point temperature of the mixture.
The preferential evaporation of the more volatile component gives rise to axial gradients of saturation temperature and of mean concentration in both phases. In a real situation, however, where the annular flow pattern is the dominant configuration, not all of the liquid is present as a film coating the inner wall of the pipe. Rather, droplets are generated from the crests of disturbance waves which travel along the liquid film and these droplets become entrained in the vapour core Figure The droplets travel at approximately the same velocity as the vapour.
All along the channel, droplets are being exchanged between the film and the core by entrainment and deposition. However, this exchange is not rapid enough to maintain equality of composition between the droplets and the film. Bearing in mind that droplet evaporation may be negligible when compared to that of the liquid film the temperature driving force in the liquid film is much higher , one can argue that in the actual situation not all of the liquid phase will be available for evaporation at a given distance along the pipe.
This hydrodynamic effect breaks down the thermodynamic equilibrium relationship existing between quality and mixture saturation temperature. For the sake of clarity, let us consider first the simplified situation in which a certain amount of liquid is entrained as droplets at the onset of annular flow and in which no further entrainment or deposition occurs downstream of this point.
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In this case, the initial film flow rate amount of liquid initially available for evaporation is equal to the total liquid flow rate less the initial entrained droplet flow rate. Even in this ideal situation where the entrained liquid flow is disregarded in the thermodynamic calculation, vapour-liquid equilibrium still demands that a fixed amount of liquid must be lost by evaporation for a given wall heat flux and, at the point of film depletion 'dryout' , the film saturation temperature must be equal to the dew point temperature at the overall composition.
Since the film flow rate is less than the total liquid flow rate, then a 'dryout' will occur at a shorter distance along the channel than it would if all the liquid were in the film, and b the film saturation temperature at a given position will be higher than the saturation temperature calculated for a case in which all the liquid flow is considered in the thermodynamic calculations.
Let us consider now the real situation where there is droplet interchange between the liquid film and the vapour core. If droplet evaporation is ignored, then it is natural to suggest that the droplets will retain a concentration equal to that of the liquid film at the point at which they were created entrained. As an axial gradient of concentration is established in the liquid film due to the preferential evaporation of the more volatile component s , droplets generated at distinct axial positions will have distinct concentrations.
Thus, at a particular position, a spectrum of concentration is found in the population of depositing droplets, these droplets having arisen from a variety of positions upstream. When calculating the rate of droplet deposition at a certain distance, one must take into account two important facts, i the difference in concentration between groups of droplets, and ii the liquid film at that distance is richer in the less volatile component than any depositing droplet. This would tend to smooth the difference between distributions of film concentration and of liquid film temperature obtained using the simplified 'no interchange' approach for each initial entrained fraction.
In the forced convective region absence of nucleate boiling , the liquid film heat transfer coefficient is primarily a function of local turbulence and of physical properties.
In general, however, the temperature change in the liquid film is dominated by that occurring in the region near the wall where the flow is laminar and turbulence is suppressed. Between this near-wall zone and the interface, the temperature changes little due to the mixing caused by turbulence. This same mixing process in the film leads to a situation where the component concentrations in the liquid film are relatively constant and the interface concentration is close to the mean fully mixed concentration in the liquid film. This result was demonstrated quantitatively by Shock , who showed that for any mixture, irrespective of the width of its boiling range, the effect of mass transfer in the liquid film is small enough to be ignored; that is, the interface concentration is close to that of the fully mixed film.
Shock also investigated the influence of mass transfer in the vapour core. He found that the effects were more significant than those for the liquid film though still small. The vapour created by evaporation at the interface has to be transported through a laminar-like layer in the vapour adjacent to the interface. This transport process is achieved by a flow normal to the interface coupled with diffusive mass transfer, the latter depending on the concentration gradient of the components.
In the work described in this lecture, these mass transfer processes have been considered in detail.
Thermodynamic Non-Equilibrium Slug Flow. The model is based on a succession of slug units consisting of a Taylor bubble formed as a result of the abrupt vapour growth in the subcooled region surrounded by a falling liquid film and a liquid slug Figure Due to the formation of the vapour plugs in the subcooled bulk region, it is postulated that the liquid slugs are initially subcooled. Additional simplifications are proposed: i a substantial portion of the energy associated with the high temperatures in the near-wall region is consumed in the process of generation of the Taylor bubble and therefore the falling film is assumed saturated, ii phase change in the liquid slug and slug body gas hold-up are negligible, iii the thickness of the liquid film surrounding the Taylor bubble is small compared with the pipe diameter, iv the mass of the liquid film is small compared with that of the slug, v phase densities are constant within the slug unit, and vi the rate of change of the Taylor bubble length with time is small compared with the ascension velocity of the Taylor bubble.
Energy balances over the slug unit and the over the liquid slug give Barbosa and Hewitt, ,. To close the model, the above equations are combined with mass conservation relationships borrowed from steady-state slug flow models Fernandes et al. The rise velocity of a Taylor bubble in a quiescent liquid, V 0 , and the falling film velocity, V LB , are obtained through empirical relationships provided by Wallis The model equations are solved so that at each step, D z a value of b defined as the ratio of the Taylor bubble length to the length of the slug unit is calculated together with the real quality, x G , and the remaining slug flow parameters.
It is postulated that the onset of slug flow is associated with the point at which vapour is initially formed and the correlation of Saha and Zuber for the Point of Net Vapour Generation NVG is used to calculate the distance from the liquid inlet up to the point of initiation of slug flow. Deindoerfer F, Humphrey A. Mass transfer from individual gas bubbles. Direct numerical simulation of mass transfer in bubbly flows. Comput Fluids ; — Direct numerical simulation of droplet formation processes under the influence of soluble surfactant mixtures.
Comput Fluids ; 93— Dumont E, Delmas H. Mass transfer enhancement of gas absorption in oil-in-water systems: a review. Eckenfelder W, Barnhart EL. The effect of organic substances on the transfer of oxygen from air bubbles in water. AIChE J ; 7: — Frossling N.