Two-Phase Flow Pressure Drop In

8/13/2019 Two-Phase Flow Pressure Drop In

1/8

Edward GrafIngersoll Dresser Pumps,

Phillipsburg, NJ 08865-2797

Sudhakar NetiLehigh University,

Bethlehem, PA 18015-3085

Two-Phase Flow Pressure Drop inRight Angle BendsGas-liquid two-phase bubbly flows in right angle bends have been studied. Numerical

predictions of the flow in right angle bends are made from first principles using anEulerian-Eulerian two-fluid model. The flow geometry includes a sufficiently long inletduct section to assure fully developed flow conditions into the bend. The strong flow

stratification encountered in these flows warrant the use of Eulerian-Eulerian descriptionof the flow, and may have implications for flow boiling in U-bends. The computationalmodel includes the finer details associated with turbulence behavior and a robust void

fraction algorithm necessary for the prediction of such a flow. The flow in the bend isstrongly affected by the centrifugal forces, and results in large void fractions at the inner

part of the bend. Numerical predictions of pressure drop for the flow with different bendradii and duct aspect ratios are presented, and are in general agreement with data in theliterature. Measurements of pressure drop for an air-water bubbly flow in a bend with anondimensional bend radius of 5.5 have also been performed, and these pressure dropmeasurements also substantiate the computations described above. In addition to theglobal pressure drop for the bend, the pressure variations across the cross section of theduct that give rise to the fluid migration (due to centrifugal forces), and stratification ofthe phases are interesting in their own right. S0098-22020001004-X

Introduction

Bubbly two-phase flows are very common in industry. Suchflows occur in numerous industrial settings including turbomach-inery, oil refineries, and nuclear reactors. Historically, due to alack of understanding, the prediction of these flows and the asso-ciated transport phenomenon have been very empirical. Thispresent work deals with the description and the prediction of suchphenomenon from first principles based on some recent successesby the authors.

The presence of the second phase enhances the transport phe-nomenon in such flows, and quite often such flows are used toimprove mixing and other processes. While the most commonoccurrence of bubbly flows is associated with boiling processes,the second gas phase is some times deliberately added to theliquid to enhance transport. The pressure drop associated withtwo-phase flows is usually larger than that in single-phase flows.As it turns out, even very small mass additions of the secondphase increase the flow pressure drop substantially. For this andother reasons, measurements and prediction of pressure drop intwo-phase flows are of great interest to the industrial community.Recent advances in computational fluid dynamics, along with theavailability of larger and faster computer facilities, have made itpossible for the numerical description of two-phase flows usingNavier-Stokes equations. The present work describes the predic-tion of two-phase flow pressure drop in ducts and right anglebends.

The flow geometry studied here includes a sufficiently longinlet duct section to assure fully developed flow conditions intothe bend. The flow in the bend is strongly affected by the centrifu-

gal forces, and results in larger void fractions at the inner part ofthe bend. The strong flow stratification encountered in these flowswarrants the use of Eulerian-Eulerian description of the flow, andthe results may have implications for flow boiling in U-bends. Thecomputed bubble trajectories through the vertical straight section,ninety-degree bend and the subsequent horizontal bend comparedvery well with high speed digital photographs that were made.However, what is more interesting is the good agreement in over-

all total pressure loss since this is typically the most difficult vari-able to accurately quantify in any numerical algorithm. The com-putational model used includes the finer details associated withturbulence behavior and a robust void fraction algorithm neces-sary for the prediction of such a flow. Numerical predictions ofpressure drop for the flow with different bend radii and duct as-pect ratios are presented. The mathematical modeling and numeri-cal technique are validated by comparison with the authors mea-surements of pressure drop as well as with data in the literature forstraight and curved duct two-phase flows. The authors measure-ments are for an air-water bubbly flow in a bend with a nondi-mensional bend radius of 5.5.

Review of Two-Phase Flow TechniquesMulti-phase flows are classified into the following flow regimes

based on the interactions between the phases: bubbly, slug, annu-lar, and dispersed flows. These regimes are essentially based onthe topological differences in the flows. Modern approaches to themodeling of two-phase flows can be divided into two broadclasses: locally homogeneous analyses and separated flow analy-ses. The majority of two-phase experimental data is for homoge-neous flows. Homogeneous flow analysis, as well as the relatedexperimental analysis, is probably not useful in understandingstratified flows of the kind in a bend. Separated flow analysisincludes Direct Numerical Simulation, Drift Flux mixture mod-els, Eulerian/Lagrangian models where the continuum is treatedby Eulerian methods and the distributed phase is treated by La-grangian methods, and the Eulerian/Eulerian two-fluid model

where both the fluids are assumed to be part of an inter-penetrating continuum. Given the processor speeds and storagecapabilities of todays computers, Eulerian/Lagrangian andEulerian/Eulerian models are the most advanced procedures thatcan be used to predict turbulent transport phenomena for the com-plex flow geometries typical of industrial problems.

Low void fraction (0.1) turbulent two-phase flows can bepredicted with an Eulerian/Lagrangian scheme the latter forbubbles with a two-equation model to describe the turbulenceNeti et al.1. Such a model includes the effect of the dispersedphase on the mean continuum properties from the Lagrangianprediction, the effect of the bubbles on the turbulence, and theeffect of turbulence on the bubbles. For larger void fractions (

Contributed by the Fluids Engineering Division for publication in the J OURNAL

OF FLUIDSENGINEERING. Manuscript received by the Fluids Engineering Division

January 6, 1998; revised manuscript received June 19, 2000. Associate Technical

Editor: M. Sommerfeld.

Copyright 2000 by ASMEJournal of Fluids Engineering DECEMBER 2000, Vol. 122 761

wnloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 04/08/2013 Terms of Use: http://asme.org/terms


2/8

0.1), Eulerian/Eulerian methods are more suitable, particularlyfor large phase density ratios and if body forces are important. Anumber of two-fluid two-phase one and two-dimensional compu-tations have recently been reported. Satyamurthy et al.2consid-ers a one-dimensional steady-state simulation of liquid metal ver-tical flows in the bubble, churn turbulent, and slug regimes.Comparisons are made with data for both void fraction profilesand total pressure drop. Graf et al. 3, have presented predictionsof bubbly flow in a driven cavity with an average void fraction of0.3 with local levels even greater. De Ming et al. 4 have pre-sented a computation of a two-dimensional bubbly flow in a sud-

den enlargement using a finite volume scheme incorporating atwo-phase modified k- turbulence model.We present here a robust algorithm for fully three-dimensional

flows and illustrative computations. Graf5 has presented detailsof the three-dimensional flow features for the basic geometriesconsidered here, including lateral void distributions and bubbletrajectories. Overall pressure loss predictions and comparisonswith data in such complex flows are discussed presently.

Computational Scheme

The salient features of the computational procedures applicablefor the Eulerian/Eulerian procedures are briefly described here andadditional details are presented extensively elsewhere Graf5.Steady, incompressible, noncondensing, two-component e.g.,water-air, turbulent, bubbly flows with no phase change are con-

sidered. Density difference between the two phases is large andthus buoyancy forces can be important. Reynolds numbers areassumed to be high and the local properties of the gaseous andliquid phases are not assumed to be the same. The differences inthe individual phase velocities, and other properties, and their ef-fect on turbulence intensity play a role in determining the overallpressure drop.

The flows in the duct and the right angle duct bend are three-dimensional in nature. The axial velocity, w is assumed to bealong z direction, the primary flow direction. The u and v com-ponents of velocity in the x and y directions are the cross planecomponents. The treatment of the equations for both the phases issimilar except for the source terms. The bubbles in the presentmodel share their space with the liquid in the same controlvolumes. As a result, the two fluids can be thought of as inter-

penetrating continua.Prediction of two-phase flows such as those described above

involves the solution of several coupled partial differential equa-tions. These include six Reynolds averaged momentum equations

three for each phase, a continuity equation for each phase, twoto four for turbulence equations for a modified isotropic k-model, and other equations for the scalars. This translates to thesolution of thousands of coupled linear equations in the dis-cretized domain.

The equations governing the turbulent flow through the 90 degbend are parabolic in nature even for fairly tight bendsHumphrey6, and thus a marching solution with two-dimensional storagefor all major variables is used. Pressure is stored as a three-dimensional variable to provide better feedback between the axialand cross plane momentum equations.

Two-phase flows with large phase density differences pose ad-ditional difficulties in computation. The lighter phase despite oc-cupying significant space and moving with a noticeably differentvelocity does not contribute much mass. Thus the second phasehas little impact on the mass continuity. Carver 7 has proposedways to minimize this problem by normalizing the equations bytheir appropriate densities, and these ideas are used in the presentcomputations.

Care has to be taken in the numerics to ensure that the com-puted void fraction is bounded 0,1. This latter condition is par-ticularly important and difficult to achieve when the primaryaxial flow is small or nonexistent. Since the mass conservationfor each finite volume and the whole domain in turn can only be

achieved to within some required tolerance usually larger thanthe gas phase contribution, unrealistically high or low void frac-tion results can be computed. This is especially serious in regionswith little flow. Even small mass residuals result in large voidchanges. The analysis of two-phase flow in a two-dimensionaldriven cavity with a moving wall Graf et al. 3 was a particu-larly good test for evaluating this phenomenon since the flowsentering and leaving the finite volumes at the core of the vortexwere rather small. The solution to this was to add an effectivemass residual term ( SP) resulting from the linearization of thesource term. This helps the procedure since then the mass residu-

als are not large enough to destabilize the void fractioncalculation.The procedures used for the prediction of the bubbly flows are

described in general terms above. The next few sections presentdetails with regard to the equations solved, turbulence-modeling,effects of bubbles on turbulence, turbulence modeling of the dis-persed phase, and issues pertaining to interfacial momentumtransfer.

Equations and Solution Details. The steady, turbulent, in-compressible, noncondensing, two-fluid, two-phase flow in theduct bend are described by a set of equations in toroidal coordi-nates. The governing equations are formulated in primitive vari-able form: i.e., the dependent velocities, u , v, w, local staticpressure, p , and void fraction, , are solved for directly. In vectorform the mass conservation, void fraction, and pressure correction

equations are

U 1

1ref

U 2

2ref

0 (1)

U 1

1ref

U 2

2ref

0 (2)

Momentum Equations are:

UUkkpkkkgMk0 (3)

where is the shear stress,g is the acceleration due to gravity,Mkis the interphase momentum per unit volume, and the subscriptk1,2, refers to the two phases. The equations for the right anglebend are in fact solved in toroidal coordinates with rectangular

grids for the cross plane representations. The generalized inter-phase momentum transfer, Mk , can include all relevant forces,including drag, lateral lift force, virtual mass, and Basset forces.The modeling of these forces is considered in a subsequentsection.

The partial differential equations described above are dis-cretized using hybrid differencing Patankar 8 on a staggeredgrid, and the resulting set of algebraic equations is solved. Thisapproach is useful in avoiding pressure-velocity decoupling in thecomputation of incompressible flows Patankar 8. The solutionprocedure consisted of solving the flow equations using the SIM-PLEC algorithm Van Doormal 9.

One of the major challenges for a two-fluid bubbly flow modelis an algorithm for computing the void fraction volume fractionof dispersed phase. Since the two phases are of very different

densities e.g., 1000:1 for water: air, the effect of the heavierfluid would totally mask any contributions from the lighter fluid.To alleviate this, the mass conservation equations are normalizedby their reference densities Carver 7. The Eulerian/Eulerianscheme needs special attention in handling several aspects of thesecond phase such as limiting of void fraction to 0,1, and otherissues related to the use of the two continuity equations. Theequation for the void fraction shown above was constructed bysubtracting the continuity equations for the two phases from oneanother. The volume fraction of the continuous phase, denoted asphase 1, was eliminated by the relation

121 (4)

762 Vol. 122, DECEMBER 2000 Transactions of the ASME



3/8

The general form of the equation being solved for the local voidfraction, 2 can be written in the form

pApeAewAwnAnsAsuAudAdS1(5)

Such an approach was found to be unstable. It violated the crite-rion that Patankar 8:

Ap Anb (6)

The coefficient ofA p originally did not meet the criterion above.

It was necessary to subtract from the terms of Ap the sum of thetwo continuity equations with each normalized by its referencedensity in a somewhat modified form. The continuity equationcontains the local void fraction. So introducing it directly wouldbring the void fraction at surrounding nodes N, S, E, W, U, andD into the coefficient for the void fraction at the point of solution( P). This would not readily permit the solution of the equation setat each node. A simplifying assumption was made for this termonly, that the local void fraction at the neighbors of P are allequal to the value at P . This is similar in logic to the approxima-tions made in the SIMPLE and SIMPLEC algorithms. With thisaddition, the criterion for the coefficients ofA P described above ismet.

The iterative solution procedure uses an initial estimate of thepressure field. The u, v, and w momentum equations for the

continuum phase and then for the distributed phase are solved.Next, the difference between the continuity equationsnormalizedby the reference densityis used to determine the correction to thepressure fieldCarver6. Theu , v, andw equations are resolveduntil a certain convergence is achieved. The sum of the continuityequations normalized by the reference densities is used to deter-mine the void fraction distribution. Finally, the equations for anytransported scalars are solved and the procedure repeated untilconvergence.

Turbulence Modeling. Previous two-fluid algorithms in theliterature have used either algebraic turbulence models or em-ployed higher-level turbulence models only for the continuousphase. The dispersed phase turbulent eddy viscosity in theseworks has often been assumed proportional to the continuousphase diffusivity i.e., a particle Prandtl number. Another ap-

proach for computing the turbulent eddy viscosity of the distrib-uted phase is to relate it to the continuous phase turbulent viscos-ity by means of an algebraic equation. The theory of a particledispersed in a turbulent flow field based on the work of Tchen10 and Peskin 11 provides such an algebraic equation. Thismodeling approach has been by Mostafa et al. 12. Turbulentflows reported in a previous workGraf and Neti 3 were com-puted both by using a constant particle Prandtl number as well asby applying a two-phase modified two-equation k- model to thedistributed phase as well as the continuous phase. While there wassome improvement noted for high void fraction simulations, ingeneral it was concluded that the additional run-times and re-sources required did not warrant the inclusion of two additionalPDEs.

In this work, a modified isotropic k- turbulence model that

accounts for the presence of the bubbles by the addition of sourceterms is used. A turbulent viscosity is computed after solving fork, the turbulent kinetic energy, and , the dissipation of kineticenergy. The proposed isotropic turbulence model becomes lessaccurate in describing complex flows where significant aniso-tropic rate-of-strain components might be presentdue to stream-line curvature or rotation or separation, etc.. Single-phase turbu-lent flow computations were performed for bends of increasinglytight radius and compared with existing data. The standard k-turbulence model yielded reasonable predictions for normalizedradius ratios as low as 2.3. Due to the segregation of the phasesthe two-phase bend that can adequately be computed with an iso-tropic turbulence model is probably not as tight as R/D2.3. It

was felt that the test geometry, i.e., R/D5.5, would be a gentleenough bend for the application of the proposed isotropic turbu-lence model. We have used various algebraic curvature correctionmodels as well as solved additional transport equations i.e., Al-gebraic Reynolds Stress Modelsto improve the quality of single-phase turbulent computations. It was felt that the first applicationof a two-phase modified turbulence model for a fully three-dimensional problem should not resort to a higher order turbu-lence model if it could rationally be avoided. Certainly more com-plex two-phase turbulence models can be tried within theframework of the present solution algorithm.

The source terms in the transport equations of the turbulencevariables arise naturally from the addition of the interphase mo-mentum term to the momentum equations. The turbulent kineticenergy transport equation is

Uk kkGSk (7)

The transport equation for i.e., the rate of dissipation ofk is

U C1

k G

C22

k S (8)

These equations allow for the possibility that the distributedphase either increase or suppress the continuous phase turbulencesince the bubbles interact with the continuous phase to both createand dissipate turbulence. The continuous phase turbulence hasbeen found to be suppressed by the presence of the dispersedphase in highly turbulent flows. Simpler models that assume thatthe total turbulence is the sum of the wall shear-induced turbu-lence and the bubble induced turbulence always result in an in-crease in turbulence.

Nondimensionalized bubble ratios between 0.04 and 0.20 wereconsidered. This range was chosen so as to study flows in whichthe bubble diameter was both smaller and larger than the integrallength scale of turbulence expected for single-phase duct flow

i.e., 0.08. The interaction of the bubbles with the turbulent ed-dies associated with duct flow are expected to result in a break-upof eddies and an overall reduction in the turbulent length scale. Atwo-phase modification of the mixing length ( ly) is suggestedAl Taeweel and Landau 13, and was also used for two-phaseflows by Neti and Mohammed 1. For a flow with quality, x , andwith bubble and hydraulic diameters, d and D , respectively, TP

is written as: TPL2 ; and L2 1(xd/D)( g/L1) .This results in a two-phase length scale that is less than it is forsingle-phase. Smaller bubbles are considered less likely to causethe break up of a turbulent eddy. The turbulent viscosity is finally

computed as tCk2/, where C1 1.44, C2 1.92, and

C(0.09) are the standard constants in the k- turbulencemodel.

Liquid Turbulence Modification Induced by Bubbles. Theinteraction of the dispersed phasebubbles/particleswith the con-tinuum phase fluid tends to modify the fluid turbulence. Wanget al.14have reported turbulence measurements in bubbly flowsand have described the effects of the bubbles on the flow. Usually,the presence of the bubbles in a flow gives rise to an increase ofturbulence. Consider the changes induced in the present curved

duct test rig when bubbles are injected into what is originally asingle-phase flow. There are strong lateral and longitudinal pres-sure gradients in the flow. These pressure gradients will generatedifferent fluid and bubble accelerations due to the mass differ-ences between the two phases. This relative acceleration betweenthe two phases can be expected to cause additional velocityfluctuations.

The effect of the dispersed phase on mean quantities of velocityand void fraction lead to theories in which the von Karman con-stant is modified by the presence of the bubbles. Hino 15 andNeti and Mohammed1 report theories in which the von Karmanconstant is modified to account for the local presence of the dis-tributed phase.

Journal of Fluids Engineering DECEMBER 2000, Vol. 122 763



4/8


5/8

the acceleration of the bubble. Using the model for the virtualmass force as given by Drew et al. 25 it can be shown that thisforce contribution can be neglected for the steady flow simula-tions presently being considered. However, for many flows thiswill not be the case, e.g., explosive flows with high spatial

accelerations.The Basset force arises from the acceleration induced changesof the viscous drag and the unsteady boundary-layer development.For the present class of problems, this force is also neglected.

Experimental Apparatus and Methods

Pressure drop measurements for a water-air bubbly flow in aright angle bend with a nondimensional bend radius of 5.5 are alsoincluded in the results presented here. The methods used for thepressure drop measurements and a description of the apparatusused are given here.

The test section was made of transparent Plexiglas and con-sisted of a 25-mm side vertical square duct followed by the 90 degbend with a horizontal square duct down stream of it. The verticalduct was 48 hydraulic diameters long, and the curved duct had a

mean radius to hydraulic diameter ratio of 5.5. The horizontal ductwas 15 hydraulic diameters long. The test section was part of atwo-phase flow loop made up of 37-mm 1.5 in. stainless steelpiping. It included a pump 1/3 HP, 10 GPM, a collection tankupstream of the pump, a water-air tank after the test section, aflow meter and the necessary flow control valves. The water flow-meter drag type float meter was calibrated by diverting and col-lecting water into a weighing tank for a measured time. Com-pressed air at the desired pressure could be injected through a setof tube manifolds into the water flow at the bottom of the verticalsquare duct. The airflow tubes had 0.5 mm diameter holes for thegeneration of the bubbles. The injected airflow was measured witha calibrated gas rotameter.

High-speed digital photographs were made throughout the testsection. This served to quantify the change in the lateral distribu-tion of bubbles throughout the three different sections of the testrig. The computed void distributions compared well with this flowvisualization even at the exit of the bend exit where the bubbles

move rapidly from the inside of the bend to the top of the hori-zontal duct Graf5.The test section with the right angle bend was instrumented

with sixteen pressure taps. A schematic of the right angle bendused in the present experiments with the pressure taps is shown inFig. 1. Pressure taps were located on the inside as well as theoutside radius of the duct bend, and care was taken to ensure thatthe taps were flush inside the duct. Pressure differences were mea-sured with a calibrated Validyne DP 15-20 reluctance type differ-ential pressure transducer. The transducer with the diaphragmused had range of 88 mm H2O with a resolution of 0.02 mm. Thetransducer output could be read with a Validyne CD379 pressuredisplay meter and was recorded with the help of a Data Transla-tion DT-2805 A/D board with 12 bit resolution and an IBM com-patible personal computer. Pressure differences were measured byconnecting the appropriate pressure taps to the transducer ports.

Due to the turbulent nature of the flow, the pressure measurementswere time variant with fluctuations. The data reported here are theresult of averaging such fluctuating data over sufficiently long100 data points in 10 second periods.

Results and Discussion

Results of the computational procedure and the experimentsdescribed above for two-phase bubbly flow in a vertical duct, aright angle bend, and a horizontal square duct are described here.Two-phase pressure drop in the ducts and the bend are shown interms of the two-phase loss multiplier, which is the ratio of two-phase pressure drop to the pressure drop that would occur if the

Fig. 1 Schematic of the right angle duct bend with pressure taps used in the ex-periments




6/8

flow were liquid only. Figures 24 show the computationally pre-dicted and measured values of two-phase loss multipliers alongwith available empirical correlations. The two-phase loss multi-plier for the flow in the right angle bend is shown as a function of

the Martinelli flow parameter, X, as presented by Chisholm andLaird 26.

X PL Pg

0.5

GLGg

2n /2

Lg

n/2

gL

0.5

(17)

This expression can be simplified ifn , the power of the Reynoldsnumber in the friction factor relation, is taken as 0.25 i.e., turbu-lent flow in smooth tubes:

XGLGG

0.875

LG

0.125

GL

0.5

(18)

where, G represents the mass flow rates, and are density anddynamic viscosity respectively, with subscripts L andg represent-ing the liquid and gas phases. Larger amounts of the lighter phaseair correspond to lower values ofX.

The computed and experimentally measured values of two-phase loss multiplier for the duct bend are compared in Fig. 2 withthe measurements of Chisholm 27 for a right angle bend in acircular pipe. Chisholms data are for R/D values of i.e.,straight pipe, 5.02, and 2.36. A curve fit of the above data cov-ering the above nondimensional pipe radii has been determined

Fig. 2. As per their choice, the square root of the two-phase losscoefficient is plotted as a function of the Martinelli parameter.Though the present experimental measurements are for a squareduct bend and Chisholms were for a circular pipe bend, thepresent pressure drop measurements seem to agree quite well with

their data. The pressure drop predictions from the present compu-tational show good agreement with Chisholms empirical data.The empirical loss correlation depends only on the amount of thegas present in the two-phase flow. Our computed pressure dropdata has at least a weak dependence on the bubble size amongother things. The computational predictions plotted are for 5-mmdiameter bubbles. Pressure losses are about 10 percent higher for1 mm bubbles, and could explain discrepancies based on bubblesizes. More detailed experiments and additional computations areneeded before more definitive statements can be made in this re-gard. The present numerical computations appear to somewhatunderestimate the pressure losses for two-phase flow in a rightangle bend.

Pressure drop in a two-phase flow consists of contributionsfrom frictional loss, acceleration effects and gravitational influ-

ence. The hydrostatic head in vertical ducts is normally notcounted as part of the pressure losses. Thus the Lockhart-Martinelli two-phase loss multiplierWallis 28 is the same forhorizontal as well as vertical pipes. The two-phase loss coefficientin the horizontal section down stream of the right angle bend isshown in Fig. 3. The horizontal duct after the bend is only about15 diameters long, and the pressure drop in this duct is probablyinfluenced by the nature of flow coming out of the bend. Themeasured pressure losses were larger than those based on theLockhart-Martinelli curve as reported by Chisholm and Laird

26. The computed pressure losses are again lower than the cor-relation but the trend appears to be correct. A comparison of thepressure loss coefficient for a vertical duct is shown in Fig. 4, andagain the computations underpredict the correlation.

The consistent under prediction of the pressure losses by the

computation could be an indication of the need for change insome basic assumptions made for the boundary conditions. Thenear wall velocityand shearare determined per the log-law wallfunction for the turbulent boundary layer. To accommodate two-phase flow effects, the mixing length ( ly ) has been modifiedusing TP . For a flow with quality, x , and with bubble and hy-draulic diameters,d andD , TP , which is written as a function ofquality, bubble and hydraulic diameters. For the bubbly flows inquestion, further modifications of the log-law may be necessary topredict the wall shear better.

The effect of void fraction on pressure drop in a right anglebend is shown more explicitly in Fig. 5 where the two-phasepressure loss coefficient is plotted as a function of void fraction.

Fig. 2 Two-phase multiplier for pressure drop versus Marti-nelli parameter for a right angle bend

Fig. 3 Two-phase multiplier for pressure drop versus Marti-nelli parameter for a horizontal duct

Fig. 4 Two-phase multiplier for pressure drop versus Marti-nelli parameter for a vertical duct




7/8

The loss coefficient is defined as: loss coefficientp/q inlet,where p is the pressure change and q inlet is the inlet dynamichead. As expected, increase in void fraction results in increasedpressure drop.

The bend radius is an important parameter in determining theflow conditions and phase segregation. Effects of bend radius onpressure drop are shown in Fig. 6. Single-phase flow typicallyshows a minimum loss at a dimensionless radius ratio i.e., R/Dbetween two and three. Diffusion losses are the major contributor

to overall losses when the radius ratio is less than two. Wallfriction losses begin to predominate for flows with radius ratiosgreater than this. For single-phase flow, the predicted pressuredrop is slightly higher than the data reported by Blevins 29. Thepresent parabolic predictive scheme is not suitable for computingflows in smaller radius ratio ducts where elliptic effects and flowseparation may be important. The two-phase loss prediction indi-cates that a minimum loss occurs for a radius ratio between fiveand six. This is higher than that found for single-phase flow. Thephase segregation that occurs with two-phase flows may be thereason for this difference. The lighter phase fills the inner part ofthe right angle bend. It is expected that this blockage will exacer-bate the increasing pressure loss that occursin single-phase flowwith decreasing radius ratio and thus tends to move the minimumto a higher value.

Flows in rectangular cross-section ducts are also of interest inmany typical industrial problems. Therefore computations werealso performed for rectangular ducts of different aspect ratios.Pressure loss coefficient as function of aspect ratio AR, i.e., theratio of the ducts height in the radial direction to its base isshown in Fig. 7. For single-phase flows, the minimum pressureloss condition was shown by Graf5 to occur for an aspect ratiobetween 1.75 and two. The two-phase minimum is predicted tooccur at an aspect ratio between one and 1.25. For smaller ductaspect ratios, the pair of counter-rotating vortices that forms insidethe bend results in an increasingly larger percentage of the totalflow being highly sheared. The loss also increases as the aspectratio increases but not as strongly as with decreasing aspect ratio.

This is possibly due to the earlier formation of multiple vortexpairs in the bend with their resulting increase in shear.

The flow in a bend results in substantial phase segregation dueto the body forces. Initially the bubbles enter the vertical ductbend with peaked values near the walls, and with symmetric dis-tribution. About half way through the bend, the bubbles are

mostly concentrated at the inner radii with the outer radii of thebend filled with liquid, see computations and digital photographsin Graf5. The bubbles end up at the outer radii top of the bendnear the exit of the bend. Such transport of the phases requiresrather complicated cross plane distributions of pressure in theduct. These cross plane pressure differences are shown in Fig. 8.The present experimental measurements of these cross plane pres-sure variations and the computed gradients are in general agree-ment. The present data show that some lateral pressure gradientexists at the inlet to the bend i.e., at the 0 deg plane. However,the present computational method does not predict this effect. Theexperimental data indicates that the maximum pressure gradientoccurs at approximately 60 deg in the bend. The computations,however, predict an increasing cross plane pressure gradient frombend inlet to exit.

Conclusions

1 Numerical predictions of two-phase bubbly flows from firstprinciples Navier-Stokes equations with minimal empirical in-put, in vertical and horizontal ducts and a right angle bend havebeen obtained. The computed overall total-pressure loss data arecompared with both the authors measurements and existing datain the literature.

2 Measurements of pressure drop for air-water bubbly flows ina right angle duct bend of dimensionless radius ratio 5.5 arereported.

Fig. 5 Variation of pressure drop in a right angle bend withvoid fraction

Fig. 6 Variation of pressure drop in a right angle bend withradius of the bend

Fig. 7 Variation of pressure drop in a right angle bend withduct aspect ratio

Fig. 8 Pressure variation in the cross section of a right anglebend




8/8

3 The pressure drop measured in the duct bend is close to thatreported by Chisholm 27 but the computations somewhatunderpredict the pressure drop.

4 The increase in pressure drop loss coefficient that occurswith void fraction is predicted well by the present algorithm.

5 The effect of duct radius ratio on two-phase pressure drop issimilar to that in a single-phase flow. However, the minimumpressure loss is predicted to occur at a higher radius ratio be-tween 5 and 6.

6 The effect of duct AR on pressure drop is similar to that seenwith single-phase flows. However, the minimum pressure loss is

expected to occur at a lower aspect ratio (1.25) than seen insingle-phase flow.7 Cross plane variations of pressure were measured and are

compared to computational predictions. The predicted level ofcross plane pressure gradient is reasonable. However, the maxi-mum in the measured pressure gradient occurring at 60 deg in thebend is predicted to occur close to the duct exit.

Nomenclature

AR aspect ratioCD drag coefficientC1 turbulence model constantC2 turbulence model constantC turbulence model constant

d bubble diameter

D hydraulic diameterFD interfacial force

G mass flow ratek turbulence kinetic energyq inlet dynamic head

R bend radiusRe Reynolds number

U velocity vectoru, v, w velocity components corresponding to x , y , z, re-

spectively

u, v fluctuating turbulence componentsVrel relative velocity between phases

X Martinelli parameterx, y , z Cartesian coordinates

x, y cross section plane of duct

z primary flow direction in ductx quality mass fraction of gas void fraction

PL friction pressure drop for liquid flowing alone in theduct

Pg friction pressure drop for gas flowing alone in theduct

turbulence kinetic energy dissipation rate

L2 two-phase multiplier

transport parameter for k or dynamic viscosity density

1ref reference density of the continuous phase

(998 kg/m3 presently2ref reference density of the distributed phase

(1.21 kg/m

3

presently shear stress kinematic viscosity

Subscripts

b bubblec centerlinei vector direction

rel relativeref reference

v vapor, lean phase1, L liquid, continuum phase2, G gas, lean, dispersed phase

References

1 Neti, S., and Mohamed, O. E. E., 1990, Numerical Simulation of TurbulentTwo-Phase Flows, Int. J. Heat Fluid Flow, 11 , pp. 204213.

2 Satyamurthy, P., Dixit, N. S., Thiyagarajan, T. K., Venkatramani, N., Quraishi,A. M., and Mushtaq, A., 1998, Two-Fluid Model Studies for High Density

Two-Phase Liquid Metal Vertical Flows, Int. J. Multiphase Flow, 24, pp.721737.

3 Graf, E., and Neti, S., 1996, Computations of Laminar and Turbulent Two-phase Flow in a Driven Cavity, ASME Fluids Eng. Div., Vol. 236, pp.

127135.

4 De Ming, W., Issa, R. I., and Gosman, A. D., 1994, Numerical Prediction ofDispersed Bubbly Flow in a Sudden Enlargement, ASME Numerical Meth-

ods in Multiphase Flows, FED-Vol. 185, pp. 141157.5 Graf, E., 1996, A Theoretical and Experimental Investigation of Two-phase

Bubbly Turbulent Flow in a Curved Duct, Ph.D. dissertation, Lehigh Uni-

versity.

6 Humphrey, J. A. C., et al. 1981, Turbulent Flow in Square Duct with StrongCurvature, J. Fluid Mech., 103, pp. 443463.

7 Carver, M. B., 1982, A Method of Limiting Intermediate Values of VolumeFraction in Iterative Two-Fluid Computations, J. Mech. Eng. Sci., 24, pp.221224.

8 Patankar, S. V., 1980, Numerical Heat Transfer and Fluid Flow , HemispherePublishing, New York.

9 Van Doormal, J. P., and Raithby, G. D., 1984, Enhancements of the SIMPLE

Method for Prediction of Incompressible Flows, Numer. Heat Transfer, 7,pp. 147163.

10 Tchen, C. M., 1947, Dissertation, Delft, Martinus Nijhoff, The Hague.11 Peskin, R. L., 1959, Some Effects of Particle-Particle and Particle-Fluid In-

teraction in Two-Phase Turbulent Flow, Ph.D. thesis, Princeton University,

Princeton, N.J.12 Mostafa, A. A., Mongia, H. C., 1987, On the Modeling of Turbulent Evapo-

rating Sprays; Eulerian versus Lagrangian Approach, Int. J. Heat Mass

Transf., 30 , pp. 25832592.

13 Al Taweel, A. M., and Landau, J., 1977, Turbulence Modulation in Two-Phase Jets, Int. J. Multiphase Flow,3 , pp. 341351.

14 Wang, S. K., Lee, S. J., Jones, O. C., and Lahey, R. T., 1987, 3-D turbulenceand phase distribution in bubbly two-phase flow, Int. J. Multiphase Flow, 13,No. 3, pp. 327343.

15 Hino, M., 1963, Turbulent Flow With Suspended Particles, Proceedings,American Society of Chemical Engineers, 89, pp. 161185.

16 Wang, D. M., Issa, R., and Gosman, A. D., 1994, Numerical Prediction ofDispersed Bubbly Flow in a Sudden Enlargement, Fluid Engineering Divi-

sion, Numerical Methods in Multiphase Flows, 185 , pp. 141148.17 De Bertodano, M. L., Lahey, Jr., R. T., and Jones, O. C., 1994, Development

of a k- Model for Bubbly Two-Phase Flow, ASME J. Fluids Eng., 116, pp.128134.

18 Besnard, D. C., Katoka, I., and Serizawa, A., 1991, Turbulence Modificationand Multiphase Turbulence Transport Modeling, Fluids Engineering Divi-

sion, Vol. 110, Turbulence Modification in Multiphase Flow, pp. 5157.19 Soo, S. L., 1967, Fluid Dynamics of Multiphase Systems, Blasisdell Publishing

Company, London, UK.

20 Jayanti, S., and Hewitt, G. F., 1991, Review of Literature on DispersedTwo-phase Flow with a View to CFD Modeling, AEA Technology Report,

AEA-APS-0099.21 Bunner, B., and Tryggvason, G., 1998, American Physical Society Division of

Fluid Dynamics 51st Annual Meeting, Nov., Paper # BA.03.

22 Saffman, P. G., 1965, The Lift on a Small Sphere in a Slow Shear Flow, J.Fluid Mech., 22, pp. 385400.

23 Saffman, P. G., 1968, Corrigendum to The Lift on a Small Sphere in a SlowShear Flow, J. Fluid Mech., 31 , p. 624.

24 Mei, R., and Klausner, J. F., 1994, Shear Lift Forces on Spherical Bubbles,Int. J. Heat Fluid Flow, 15 , pp. 6265.

25 Drew, D., Cheng, L., and Lahey, Jr., R. T., 1979, The Analysis of VirtualMass Effects in Two-Phase Flow, Int. J. Multiphase Flow,5 , pp. 233242.

26 Chisholm, D., and Laird, D. K., 1958, Two-Phase Flow in Rough Tubes,Trans. ASME, 80 , pp. 276286.

27 Chisholm, D., 1980, Two-Phase Flow in Bends, Int. J. Multiphase Flow, 6,pp. 363367.

28 Wallis, G. B., 1969, One-Dimensional Two-Phase Flow, McGraw-Hill, NewYork.

29 Blevins, R. D., 1984, Applied Fluid Dynamics Handbook, Van Nostrand Re-inhold Co., New York.


Two-Phase Flow Pressure Drop In

Documents

Transcript of Two-Phase Flow Pressure Drop In