# Theoretical Work On Liquid Liquid Dispersed Flow Biology Essay

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Liquid-liquid two-phase flows are often encountered in many industrial areas, like chemical, pharmaceutical, foodstuff, oil industry, and in particular as regards oil industry is underlined its transport ( pressure drop ) and as regards chemical applications it is important to study hold-up, drop size and fluid properties, or in academic research where mathematics models and experimental correlations are developed. However, despite a lot of applications, this flows are not so studied as gas-liquid systems because in liquid-liquid systems density different between the two phases is low but could be an important viscosity ratio, which can extends over a range of many order of magnitude. So it is more difficult to determined fluids properties and to study their behaviour, for example different emulsions can show both a Newtonian or non Newtonian behaviour in relationship with phases flow rates. So it is important to define the different flow regimes and build flow pattern maps ( see paragraph 1.7 ) where four basic flow regimes are classified with visual observations: stratified flow, slug flow, dispersed flow and annular flow, where one of the liquids forms the core and the other the annulus, but in many case there are a combinations of these regimes, that changes in function of fluids properties ( density and viscosity) and flow rates ( see fig. xxx ).

1.2 Brauner description of dispersed flow

The objects of this work are liquid-liquid two phase flow systems and in particular the dispersion of two immiscible liquids, where one of the two liquid is the continuous phase and the other is dispersed in it; if the two liquids considered are water and oil there are water in oil ( w/o) and oil in water ( o/w) dispersion. Also a stable dispersion of fine droplets is called emulsion that is generated with the presence of surfactants that inhibit coalescence of droplets. However is focused attention on o/w dispersion in pipes and is followed Brauner ( 2004 ) description of the phenomena.

So a dispersion can be considered as a pseudo-homogeneous fluid which have mixture properties evaluated from the properties of continuous and dispersed phases; as regards dilute dispersions ere used some simple equation for mixture properties but as to dispersions which have a more significant percentage of dispersed flow it is necessary to consider other parameters such as slip velocity ( different between the velocity of the continuous phase and the dispersed one caused by superficial tension of drops and viscosity ), drift velocity ( a contribute given by the density different of the two phases ) and bubbles coalescence; consequently gravity field effects and interactions between the two liquid phases became important.

At first is defined mixture density that is calculated started from the hold-up of the dispersed phase . Then as regarding the dynamic viscosity ,when the slippage between the two phases is significant it is takes as the viscosity of the continuous phase , and for dilute dispersion it is used Einstein's relation

( 1906 ):

can be underlined that mixture viscosity it is affected mainly by the viscosity of the continuous phase and increases with the increasing of Îµd ; in addition to it exist other models or correlations purposed in literature in form of such as Ball and Richmond ( 1980 ) and Toda and Furuse ( 2006) correlations:

however the previsions of these models are not so different from the Einstein's model prevision ( Grassi et al. 2008 ) , consequently Einstein's correlation is used to predict mixture viscosity not only for very dilute dispersion but in general for more dense dispersion. Also in Guet et al. (2006) are purposed other relations for effective viscosity, specifying the condition in which the can be used:

valid for higher concentrations ( Brauner, 1998 );

valid for dispersed fractions of up to Îµdâ‰ˆ0.1 (Becher, 2001);

an exponential law in which k is a fitting parameter ( k=2.5 for o/w dispersion) ;

a linear average model ( Elseth, 2001 );

Finally as for the dependence from the temperature the viscosity decreases with increasing the temperature, , where T is the absolute temperature and A, B are constants that depends form the type of fluid and shear rate.

1.2 Dimensionless numbers

Next the definition of mixture properties for the equivalent homogeneous fluid can be defined some dimensionless numbers, based on them, that appears in many correlations for pressure drop and drops size.

First of all mixture Reynolds and Weber numbers:

where D is the diameter of the pipe, Ï„ is the shear stress, Ïƒ is the superficial tension between the two fluids.

Then ,when multiphase systems are considered, is suggested in literature the importance of Eotvos number that is the ratio of gravitational to surface tension forces, defined as

where âˆ†Ï is the density difference between the two fluids ( continuous and dispersed phase ), g is the gravitational constant, D is the diameter of the pipe and is Ïƒ the superficial interfacial tension. In particular low Eo and high Eo systems have different behaviour in terms of flow pattern maps, transition boundaries and regime occurrences.

1.3 Hold-up models

Hold-up it is an important property to find for a two-phase flow system: it indicate the effective cross section area that a single phase occupies on the pipe; it is useful to know hold-up in order to establish the real quantity of each phase that flows and this aspect have a lot of applications in oil and chemical industry.

Hold-up is defined in terms of superficial ( Uis ) and phase velocity (Ui ):

where Qi is the i-phase flow rate, Îµi is the hold-up of the i-phase, Atot and Ai are respectively all the cross section area of the pipe and the cross sectional area of the pipe occupied by the i-phase.

The simplest approach is the homogeneous model which neglected a possible difference between the phase velocity of the two liquid phase; so the dispersed phase have the same velocity of the continuous one and slip velocity ( the different between the two phase velocity ) is not considered. Hold-up can be determined by only the flow rate of the two liquids:

In addition to it, specially when water forms the continuous phase and flow velocity is low , homogeneous model can't be used, because due to density difference dispersed phase tend to move at different velocity than the continuous one: in particular can be observed large droplets ( such as oil bubbles ) that present a significant slippage. In this case is used Zuber-Findlay (1965) drift flux model in which the phase velocity of the dispersed phase is expressed in terms of mixture velocity Um and drift velocity ud ( a contribute caused by buoyancy effects ):

where Co is a distribution parameter ( Co=1 for uniform droplets concentration, Co>1 when the droplets tend to flow at the centre of the pipe and Co<1 when the concentration is higher near the wall ). In order to estimate ud in Brauner (2004) is purposed this equation:

nd=1.5÷2.5 for liquid-liquid dispersion ( Hassan and Kabir 1990 and Flores et al. 1997 ), uâˆž is the terminal rise velocity of a single droplet in the continuous phase and correlations for it can be founded in Clif et. al (1978) or, for distorted drops, in Harmathy (1960) is presented this relation:

where there is no dependence from the drop diameter.So hold-up calculated with the drift flux model is expressed in this algebraic equation valid for concurrent and counter current flows:

where Î² is the inclination angle and Î² > 0 for downward inclination.

1.4 Models for pressure drop predictions

For fully developed flow, the total pressure gradient, dP/dz ,calculated from the homogeneous model, is the sum of the frictional pressure gradient, dPf /dz and the gravitational pressure gradient, dPg/dz

where the z coordinate is attached to the direction of the continuous phase flow, Î² > 0 for upward inclination, D is the diameter of the pipe, Ïm , Um are respectively the mixture density and velocity, fm is the mixture friction factor based on mixture Reynolds number. Mixture friction factor can be evaluated applying single phase flow equations ( dispersion have a Newtonian behaviour for dilute and modest dispersed phase concentration, it is possible to speak about dispersion as a non Newtonian fluid only for very dense dispersion ) : for laminar flow can be used Haegen-Poiseuille equation, for turbulent flow can be used Moody diagram or Blasius correlation for smooth tubes or other correlation suggested in literature ( such as Colebrook ( 1939) equation yields ).

The analytic friction factor for laminar flows

Blausius correlation

1.5 Drops size: K-model

Drops size and their characteristics are important aspects for analysing the hydrodynamic and transport phenomena in dispersed flow: breakup mechanism, coalescence, drops interaction are influenced by the flow velocity field, surface forces, collisions and some other additional factors.

So it is important to predict the size of bubbles or drops and in particular in turbulent flow ( the common type of flow field that can be observed in dispersions in pipes ) and two models are available: H-model ( Hinze's model 1955) and K-model ( Hunghmark's model 1971). K-model in particular is valid when the viscosity of the dispersed phase is much larger then that of the continuous phase: if Î¼d /Î¼c >>1 viscous forces became important and can't be neglected.

The model suggested that the maximal drop size dmax can be evaluated as the maximum of two ideal maximum drops diameter obtained from a static force balance between the eddy dynamic pressure and the counteracted surface tension force ( considering a single drop in a turbulent field ) (dmax)o , and from a local energy balance ( specially for dense dispersion in which coalescence is very significant )(dmax)Îµ. As a result K-model can be express in these equations for dmax>0.1D ( Brauner 2004 ):

where D, Îµd are respectively the diameter of the pipe and the hold-up of the dispersed phase; Ck=O(1). The dimensionless critic Weber and Reynolds numbers of the continuous phase are defined as :

where Ï„ is the shear stress, Ïƒ is the superficial tension between the two fluids, Ïm is the mixture density and Î¼m is the mixture dynamic viscosity ( calculated for example with Einstein correlation ).

1.6 Flow pattern maps: o/w dispersion area

An appropriate point to study multiphase flows is a phenomenological description of the geometric distributions that are observed, and a particular type of geometric distribution of the fluids is called a flow pattern. Flow patterns are recognized by visual inspection and the results are displayed in the form of a flow pattern map that identifies the flow regime in different areas of a graph ( on the axes there are the superficial velocity of the two fluids ). As regards the boundaries between the various flow patterns in a flow pattern map, they occur because a regime becomes unstable and in Brauner (2004) are purposed models to predict them and experimental results. Also in Grassi et al. (2008) is presented an experimental validation for boundaries prediction in two-phase high-viscosity ratio liquid-liquid flows ad is reported in Fig. XXX the flow pattern map obtained.

The area occupied by o\w dispersion is highlighted and can be observed in a wide range of superficial velocities: at low oil superficial velocity there are dilute dispersions and at higher oil superficial velocity there are more concentrated dispersions ( Fig. XXX ). But it is necessary to consider carefully the map region comprised between 0.03m/s and 0.2m/s oil superficial velocities, because ,there, is expected "bubbly flow" that is quite different from oil in water dispersion; in fact there is bubbly flow where larger oil drops is concentrated in the upper part of the pipe under the effects of coalescence and gravitational force. The differences from the two flow regimes can be observed clearly in Fig XXX, where on the left is reported the transition, at a fixed oil superficial velocity, between dilute o/w dispersion and a bubbly flow ( drops became larger and occupies the upper part of the tube and are evident coalescence ), and on the right the transition between more dense o/w dispersions to bubbly flow to slug flow ( intended as the alternation of elongated big drops and water cells in the upper part of the tube ).

So in this work is considered o/w dispersion in the highlighted region, but not bubbly flow, because the two flow pattern have a quite different behavior.

2.1 Experimental set-up description

All the experiments are performed in a experimental facility ,which has been built at the University of Brescia, shown in Fig.xxx in order to simulate the flow of two immiscible liquids in a pipe; in these case oil and water are used and they are initially stored in 1m3 tanks. As regards two fluids physical properties oil has an 886 kg/m3 density and 0.9 Pa s dynamic viscosity, water has 1000 kg/m3 density and 0.0013 Pa s dynamic viscosity and the superficial tension Ïƒ and the viscosity ratio Î¼d /Î¼c are respectively 0.05 Nm-1 and 692.31. Two fluids are moved from their tank storage with a water and oil pump ,while oil and water flow rates are measured by a turbine and a magnetic flow-meters respectively, until the injection device. In addition to it oil screw-type pump is controlled by a mechanical reducer, water is supplied by a centrifugal pump connected to a frequency inverter that assures control of the pumped flow-rate. The injection device has been studied to promote the formation of core annular flow ( see Grassi et al. 2008) but it is able to generate good dispersion: oil enters via the central tube, while water is introduces as an annulus between the central tube and the glass pipe, then at the exit the two phase mixture reach the test pipe. In fact the injection device is connected with a 9 m long glass pipe with an internal diameter D=0.022m ( L/D=428) on which is placed the pressure drop (PD), on a 1.5 m tract, and a hold-up transducer (CP) ( see Fig xxx). At the end of the pipe there are the Quick Closing Valves ( QVC) and finally the end pipe is at atmospheric pressure: oil and water mixture is collected in a 1m3 tank, where the two fluids are separated as a consequence of density difference ( oil occupies the upper part of the tank and water the lower one ) and they are pumped to their storage tanks. Also long the pipe there is a glass box inserted to reduce optical distortion in order to allow a correct observation of the flow by the observation instrument.

2.2 Capacitive probe for hold-up measure description

In this work a capacitance sensor for hold-up measurements is used: it was developed in the Brescia University , see Demori et al. (2009), in which is presented the probe for core annular flows. To tell the truth other kind of sensors, like nuclear sensors, have very high sensitivity and are more suitable for dispersion investigations but they are much expensive, capacitive sensors instead are relatively cheap and easy to install ( they are mounted on the external part of the pipe) but as can be seen in literature are used only with non conductive fluids. So the aim of the probe used in this experiments is to work with conductive fluids and to detect small amounts of the dispersed phase ( hold-up measured with o/w dispersion are smaller then hold-up with core-annular flows: consequently a good sensitivity is needed). The working of the probe based on the estimate of hold-up variations, measuring the variations of the electric properties of the fluids in the pipe: it is important to establish where electrodes have to be located and how flow regime is linked with the distributions of electric properties.

First of all, to start the probe description, it can be divided in tree principal parts as shown in Fig. XXX :the measurement head with sensing and guarding electrodes placed on the pipe ,the electronic circuit and the data acquisition system.

As regards the measurements head, starting from the results of Abouelwafa & Kendall (1980) and Canière et al. (2008) the concave configuration is chosen where the electrodes that are directly placed facing each other on the external surface of the pipe and the guarding electrodes are placed before and after sensing electrodes. Also is important to underline the fact that he probe need high sensitivity because the continuous phase near the pipe wall ( a conductive fluid, water in o/w dispersions) forma sort of shield for variations of dispersed phase properties.

Then about the electrode design there is a covering angle Î¸e=90Â°, because as shown in Demori et al. (2008) as the opening angle decreases, measured capacitance become less sensitive to the spatial resolution of the fluids, and it is 0.1 m long ( LM=m), a compromise to make the measure quite local and not in contrast with electrode covering angle.

In addition to it as it is shown in Demori et al. ( 2009), when there is a conductive fluids on the pipe, in order to control the electric field and to minimise the influence on the measurements due to the current losses are used guard electrodes ( LG=0.4m).

In fact the phenomena has not only to be considered only in the cross section but also along the pipe ( see Fig.xxx); so the measurements is influenced by external couplings that can be quantified in a loss current IL which leak out from the measurements section to the axial direction along the pipe. Two hypothesis are made ( Demori et al. 2009) about these external couplings: the contact between the conductive water wire with the metallic parts of the hydraulic set-up ( like pumps ) or the contact with the metallic parts of the whole experimental set-up, like supports or mounting fixtures.

So the electrical system can be built considering the current losses with the aim to correct the output signal from the effects of external couplings.

The electronic circuit work at a working frequency fp of 2 MHz because increasing fp the sensitivity increases and the shielding effects of the conductive fluid decrease; with reference to Fig. XXX the guards electrode are connected to the terminal GND, which is grounded, the measurement electrode to terminal M and the other electrode to terminal P, then the circuit excites terminal P with a sinusoidal voltage. Therefore all these components are linked with the transimpedance amplifier ( an inverting-configuration operational amplifier ) and , consequently, with the demodulation stage; finally the output voltage Vmeas ( DC ) is cached by a data acquisition and visualisation system based on a 16-bit A/D data acquisition board ( PCI-6259) connected to a personal computer. All the probe is set to work in negative sensitivity, so the condition with only water flowing gives an output of 0 V ( reference value ) and the addition of the dispersed phase can be observed as increase of Vmeas , because generally the dispersed phase has a lower dielectric constant respect the water and as a consequence the rising of dispersed phase hold-up gives a decrease of measured capacitance C.

After all the sensitivity of the whole probe is 2V/pF and the output voltage variations for dispersions are in order of hundreds of mV.

2.3 Capacitive probe for hold-up calibration curve ( FEM model )

The calibration curved for dispersed flow ( o/w dispersion) is obtained considering the two phases in the pipe as a homogeneous equivalent fluid with opportunely weighted electrical properties ( permittivity and conductivity ). As suggested in Geraets & Borst (1988), in this work it is used Maxwell (1881) model:

where Îµr,i is the relative permittivity of the i-phase, Ïƒ is the conductivity ( S/m) and Hd is the hold-up of the dispersed phase. This model is valid until the drops of the dispersed phase are so small that their interaction can be neglected, so it is valid only until hold-up Hd < 0.2, and in particular gives good results for Hd < 0.1.

Once assumed Maxwell model the equivalent electric circuit for dispersed flow is shown in Fig XXX where Ct describes pipe behaviour and Rfl,eq and Cfl,eq represented the equivalent behaviour of equivalent fluid. In this experimental set-up Ïƒw=0.03 S/m, Îµr,w =78, Îµr,o=2.7.

So the calibration curve for the probe is obtained with a FEM simulation using COSMOL MULTIPHYSICS in terms of RCD (see Fig. xxx) . RCD stays as "Relative Capacitance Variations" and it is useful to define in this way the changing of oil and water electric properties; it is defined as

where C is measured capacitance, Co and Cw are the capacitance when oil and water respectively flow in the pipe. First of all it is important to underline that 20% variation of measured oil hold-up corresponds to a very small variation of RCD value

in order of O(0.02); this fact means that it is necessary to make the experimental data acquisition with a lot of attention.