A Preliminary Numerical Approach for the Study ofCompressed Air Injection in Inverted Siphons

A. Freire Diogo1 and Maria C. Oliveira2

Abstract: Inverted siphons are particular structures in sewerage and drainage systems that are frequently associated with difficulties inhydraulic design and operating conditions. Air injection in the base of the rising branch of the sanitary siphons may solve problems oftenfound in conventional design, such as the available hydraulic head and the flow velocity required, or the frequent deposition of suspendedsolids and organic matter. A model was developed for numerical integration of the energy differential equation for gradually varied steady flowof the two-phase flow of the rising leg considering isothermal conditions. The results of the model application were compared with publisheddata measured on air-lift pumps and with data obtained in a siphon experimental setup with two barrels of different diameters, which wasspecifically constructed and tested for the research. The numerical results for the set of parameters used and corresponding calibration datashow a good agreement with the experimental measurements, primarily for the barrel with the smaller diameter. DOI: 10.1061/(ASCE)HY.1943-7900.0000702. © 2013 American Society of Civil Engineers.

CE Database subject headings: Two phase flow; Flow resistance; Steady flow; Turbulent flow; Siphons; Numerical analysis.

Author keywords: Two-phase flow; Fluid fraction; Void fraction; Energy equation; Flow resistance; Head loss; Isothermal conditions;Flow pattern; Steady flow; Turbulent flow.

Introduction

Because water is indispensable to life, all humans and many humanactivities use water and produce wastewater that includes sus-pended solids and organic matter, usually associated with pollutantcharacteristics including pathogens, metals, sulfates, methane andchlorides, or nutrients. Systems designed for collection, transpor-tation, pumping, and treating wastewater are fundamental elementsof urban infrastructures. The networks generally operate predomi-nantly by gravity and as open-channel flow (i.e., the flow occurs inthe closed section at partial depth, or with a free surface) to avoidseptic conditions, which is crucial for the operation of the systemand the integrity of the infrastructure, which can be attacked bycorrosive effects.

The adoption of large integrated systems instead of the prolif-eration of small isolated networks, serving by small partial drainagebasins, may present important economy scales and an easier controlof environmental issues, either in the development of new infra-structure or the expansion or remodeling of existing infrastructure(Diogo 1996). However, this may introduce new problems andchallenges (Diogo and Graveto 2006). With growing systems,the travel time and probability of septic conditions increase, newpartial drainage basins must be included, and topographical diffi-culties and obstacles are generally substantially increased.

In sanitary sewer networks, inverted siphons are complementaryinstallations of sewers, which operate by gravity and pressure pipeflow. They were developed for crossing obstacles such as valleys,streams, lakes, or unleveled passageways, and are comprised oflowered U-shaped conduits, with maximum downstream levelrecovering. With an inlet and outlet chamber and frequently twoor more barrels in parallel between them, inverted siphons representvery simple and attractive solutions, principally because of the rel-atively low investment cost. They may also have considerable tech-nical and economic advantages in relation to the more commonlyavailable alternatives, e.g., aerial crossings by host or dedicatedbridges, channel-bridge structures, or even wastewater pumpingsystems.

However, in conventional siphon design (e.g., ASCE 1969,1982; MetCalf and Eddy 1981), acceptable values (required for de-tention time, self-cleaning velocities, and other design parameters)are frequently difficult to verify, principally in the case of largelengths, deep depressions, steep slopes of the rising leg, and fre-quent variability of water inflows. This incapacity causes frequentoperational problems due to the absence of airing for long timeintervals with very low flow rates, potentially up to hours, withsubsequent sulfide generation and the possibility of frequent block-ages that are sometimes difficult to eliminate. Engineers frequentlybalance and mitigate several design alternatives and are many timesinclined to find other layout solutions.

A large sewage inverted siphon of several hundred meters and aconsiderable rising leg for the crossing of a significant valley ofapproximately 40 m depth in the south of Santiago Island, CapeVerde, with an air injection solution, was preliminarily conjecturedin Diogo (2008) and Diogo and Oliveira (2008). It was assumedthat with compressed air injection in the base of the rising leg,the aeration and oxygen that the wastewater lacks would be sup-plied, the self-cleaning velocity would be increased, and the solidtransport would be significantly assisted due to the flotation phe-nomena (related to the air-lift effect).

1Assistant Professor, Dept. of Civil Engineering, Univ. of Coimbra,Pólo II, 3030-290 Coimbra, Portugal (corresponding author). E-mail:afdiogo@dec.uc.pt

2Former Student, Dept. of Civil Engineering, Univ. of Coimbra, Pólo II,3030-290 Coimbra, Portugal.

Note. This manuscript was submitted on April 2, 2012; approved onNovember 8, 2012; published online on November 10, 2012. Discussionperiod open until December 1, 2013; separate discussions must be sub-mitted for individual papers. This paper is part of the Journal of HydraulicEngineering, Vol. 139, No. 7, July 1, 2013. © ASCE, ISSN 0733-9429/2013/7-772-784/$25.00.

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This solution, to the best of the authors’ knowledge, has neverbeen investigated nor established in the technical literature, andconsequently there are no specific data. However, the applicationof the air-lift technique in inverted siphons seems to solve the tradi-tional difficulties that are frequently encountered in the design andoperation of siphons, improving the hydraulic performance and theexploration conditions.

Between multiple issues and nuances concerning the siphondesign, details of installations, wastewater aeration, and solid trans-port, two basic questions arise. What air flow rate should beinjected? With what frequency should the air be supplied in anacceptable mode, or in optimal conditions? Analytical and exper-imental studies (Oliveira 2009; Diogo and Gomes 2011; Diogo andApóstolo 2012) were advanced in the Laboratory of Hydraulics,Water Resources, and Environment (LHRHA) of Coimbra Univer-sity, and a research project was initiated.

A preliminary numerical model was developed for the inte-gration, by finite-difference, of the energy differential equation(Bernoulli differential equation) for two-phase steady flow in therising leg considering isothermal conditions. The model equili-brates the energy at the air injection point between the siphonsingle-phase flow of the descending leg and the two-phase flow ofthe rising leg, based on the time-average fluid fraction of the mix-ture inside the pipe. It allows, as a first approach, quantification ofsome of the advantages expected for inverted siphons using thetechnique. For each air flow rate, several parameters were studied,as follows: (1) the allowable hydraulic head gain for a fixed waterflow rate; (2) the increase of water flow rate, for a fixed availablehydraulic head; and (3) the design diameter decrease and sub-sequent mean velocity rise, maintaining the remaining variables.

Themodel was first applied and comparedwith experimental datapublished in the literature, obtained for air-lift pumps (Yoshinaga andSato 1996;Khalil et al. 1999;Kassab et al. 2009), and subsequently todata measured in a preliminary siphon experimental installation,which was constructed in the LHRHA. This paper presents thedeveloped model and the corresponding results.

Artificial aeration is traditionally and extensively used in thearea of wastewater engineering to accomplish biological treatmentprocesses, such as in systems of aeration tanks for organic pollutionremoval or nitrification, aerated lagoons, oxidation ditches, or high-purity oxygen systems (Mueller et al. 2002) in wastewater-treatmentplants (WWTPs), and sometimes in force mains of pumping sys-tems or critical main sewers when natural aeration is insufficient(Lima Neto et al. 2007). The preservation of aerobic conditions,with acceptable dissolved oxygen rates, can be vital for the explo-ration conditions and the system integrity, and for the quality of thetreatment and of the entire surrounding environment. Based on anexperimental study in a 38-mm pipe for aquaculture applications,Reinemann and Timmons (1989) concluded that the oxygen transferefficiencies for the more common flow patterns of air-lift pumps isof the same magnitude or even higher than in the diffused aerationsystems. The ability of the pumps that work by air emulsion forlifting water and carrying solids, particles, and slurries is well doc-umented and is used in many applications, such as pumping in wellsor mines, well drilling (Nenes et al. 1996), the chemical industry(Yoshinaga and Sato 1996), grit removal, dredging (in lakes, estua-ries, or harbors), and mineral extraction (Khalil et al. 1999).

Two-Phase Flow and Air-Lift Pumps: Review ofFundamental Aspects

Two-phase flows currently have a wide range of field applicationsand topics of research, with very different fluids and geometries,

since condensers, refrigeration systems or minichannels (Dutkowski2009), until large-diameter bubbly columns, circulation tanks, andair-lift reactors (Mudde 2005). Alongside experimental research,which typically requires great accuracy in flow metering [usuallywithin than 5% (Oddie and Pearson 2004)], computationalfluid dynamics (CFD) is increasingly used in two-phase flows(e.g., Bhramara et al. 2008; Moraveji et al. 2011), and particularlyin air-lift pumps (Pougatch and Salcudean 2008). According toMudde (2005), in the field of aeration and recirculation, bubblycolumns and air-lift reactors (widely used in WWTP, for instance)have been thoroughly investigated in the preceding decades, anddespite refinement of the experimental techniques, thoroughobservation of local phenomena, computational resources, compu-tational techniques, and the enormous amount of literature, theunderstanding of bubble flows and dense swarms of bubbles isnot yet fully conclusive.

The fundamental aspects related to air-water two-phase flowsand the developed model presented herein are the flow patternsand characteristics, the velocity of the air and of the water in themixture, the fluid and the air (or void) fractions, and the resistancelaw of the two-phase flow. Most of the previous theoretical andexperimental research available in literature, in the general contextof the present paper, was developed for vertical pipes and appliedto air-lift pumps. There are currently many scientific papers devotedto air-lift pumps in many different areas. Reinemann (1987),Yoshinaga and Sato (1996), Kassab et al. (2009), Nicklin et al.(1962), Kato et al. (1975), and Clark and Dabolt (1986) are exam-ples. A rigorous general theory does not seem to exist, and theprimary results, details, and performance are frequently comple-mented experimentally with pump testing. In this paper, only themore important aspects directly related to the developed modelare examined. An air-lift pump is a vertical pipe partially filled witha fluid, frequently partially immersed in a water feeder tank or waterbody (e.g., as in a mine or a lake), and submitted to air injection,typically in a section that is close to its downstream end. Geomet-rically, it can beviewed as a particular case of an inverted siphonwitha negative available maximum hydraulic gradient (the pump lifts afluid or amixturewith solids, grits, or slurries), a short or null verticaldescending leg (flowing upward), and a vertical ascendant pipe.

The flow patterns that are present in a vertical pipe are basicallydependent on the mean velocities of the water and air, when flowingalone, and are described and mapped in Taitel et al. (1980). Withincreasing air flow rate, the air or vapor progressively aggregatesand preferentially occupies the central zone of the closed tube.The flows are observed and classified, in sequence, as bubble (char-acterized by multiple dispersed bubbles), slug (with large spaced-bubbles in ballistic shape until near the wall), churn (when the airflow more disordered detaches from the pipe), and annular (thewater flow forms a ring or film adjacent to pipe wall). The flowproprieties and behavior are closely dependent on the flow patternthat is reached. For an inverted siphon it is generally expected that acomparatively weaker relation between the air and water flow ratesexists than the relationships that occur in a pump. This is becausethere is no need for static lift, and thus only the bubble, slug, orchurn patterns are expected to occur. Also, the rising leg is gener-ally inclined, and the air tends to deviate from the central zone.Depending on the velocity, it tends to occupy more or less markedlythe upper portion of the pipe section.

Because water flows (in ordinary uses) approximately at con-stant temperature, it is frequently assumed that air flow in two-phase mixture occurs isothermally. As in common single-phasescenarios of hydraulic steady flow, water behaves as if it is incom-pressible, and air-lift pumps are typically calculated on bases of twoequations: the equation of motion and the equation of continuity.

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Most existing research was performed by applying the momentumequation and, occasionally, the energy equation for the verticalpump as a whole. Because of the typical low relative variation ofthe absolute pressure that occurs between the inlet and outletof the pump, air flow is also frequently considered as incompress-ible or approximately calculated, sometimes using an averagevalue. As examples of exceptions, Nicklin (1963) using the energyequation, identified the requirement for an integration and proposeda graphical approach, and Nenes et al. (1996) used an unsteady setof differential equations, based on the momentum equation, solvedby a numeric integration, and an interphase friction approach.

It is generally accepted for all flow patterns that the air meanvelocity (Ug) rising in a circular vertical tube is greater than thecorresponding average velocity of the air-water mixture, Umixt ¼ðQf þQgÞ=A, where Qf and Qg = fluid and air flow rates, respec-tively; and A = cross section of diameter D. A drift or slip betweenthe two phases occurs, and the larger the difference of velocities,the larger the loss of pump efficiency. Several approaches and em-pirical correlations for this phenomenon or the resultant void frac-tion have been proposed. Nicklin et al. (1962) and Nicklin (1963),based on experiments in single bubbles, proposed for Ug in slugflow an approach that can be expressed in a generic form:

Ug ¼ CQf þQg

Aþ Ugb ð1Þ

in which C is an experimentally determined coefficient; and Ugb =upward velocity of the bubble when the fluid is not moving (usuallyknown as a Taylor bubble). According to those authors, Ugb wasconfirmed to be approximately equal to 0.35

ffiffiffiffiffiffigD

p, where g =

gravitational acceleration, and C was found to be about 1.2 forReynolds numbers of the fluid slug greater than 8,000. The coef-ficient C, also known as distribution parameter (Hibiki and Ishii2003; Shen et al. 2010), approximates itself from the quotientbetween the maximum and average velocity in a water turbulentflow (the ratio is exactly 60=49 when the velocity profile in thesection can be expressed by the seventh root law).

Reinemann et al. (1990), for small-diameter air-lift pumps andsequences of gas slugs, obtained experimentally a coefficient ofaround 1.2, even for Reynolds numbers of the mixture, given byR ¼ UmixtD

νf, where νf = fluid kinematic viscosity, as low as

1,000, or even 500, which was explained by the vortices and tur-bulence generated in the liquid in between and by successivebubbles. Clark and Dabolt (1986) considered the impossibilityof a general value for C but, based on examination of previouslypublished data, including for large-diameter conduits, proposed 1.2as the best approximation for the slug flow. For all bubble flows, asimilar consensus for the distribution parameter to be used defini-tively does not seem to exist. Nicklin et al. (1962), for example,starting from a very simple hypothesis (the assumption of smallbubbles that are well-distributed over the cross section), advanceda value of 1.0 and recognized that a change of Ugb may be alsonecessary, according to the specificity of the flow pattern andparticularly the vicinity between bubbles.

Based on a large quantity of experimental data on single airbubbles (with diameter d between 15 and 79 mm) in vertical watercolumns with null net flow (4 and 6 m in height and diametersbetween 51 and 630 mm), Krishna et al. (1999) confirmed theDavies-Taylor-Collins correlation. This correlation indicates thatthe bubble rise velocity varies logarithmically between 0.25

ffiffiffiffiffiffigD

pfor d

D ¼ 0.125, and 0.35ffiffiffiffiffiffigD

pfor d=D > 0.6 (the coefficients are

0.33 and 0.34 for d=D ¼ 0.3 and 0.4, respectively). For a swarmof large bubbles, Krishna et al. (1999) claimed an extraordinaryvelocity increase relatively to isolated bubbles. The two velocities

are related through an acceleration factor that was found approx-imately between 3 and 6. For bubbly vertical jets in a water tankwith a square transversal area of 1.20 m on a side and 0.76 mheight, starting from the bottom of the tank with nozzles of diam-eters between 4 and 13.5 mm, Lima Neto et al. (2008) observed agreater bubble slip velocity than the superficial velocity for isolatedbubbles. Simonnet et al. (2007), in a prismatic tank with sides of0.1 m and 1 m in height, analyzed the relative velocity of a swarmof bubbles versus the local void fraction. Unlike bubbles that weresmaller than 5–7 mm in diameter, for larger bubble diameters andlocal void fractions greater than 15%, an increase of the relativevelocity of the bubbles was detected.

The geometry, and especially the ratio between the pipe verticallength of the two-phase flow (L) and the internal diameter, seems tohave a determinant role in the flow patterns reached and motionalbehavior for a given air and water superficial velocity of flow, par-ticularly for larger sections (see Ohnuki and Akimoto 1996, 2000).Several approximate correlations for the distribution coefficient (C)and drift velocity (Ugb) have been proposed for large-diameterpipes (Hibiki and Ishii 2003; Shen et al. 2010). Reinemann andTimmons (1989) stated that air-lifts with a low ratio between lengthand diameter (L=D < 50) present a considerable efficiency lossbecause of a great increase of the slip between the phases, witha severe increase of the air average velocity for L=D < 15. Theyassumed that Ug could be dependent on L=D for a given flowpattern, mode of air injection, and void fraction range.

The air and fluid fractions in the pipe (fg and ff , respectively),and consequently the average density of the mixture (ρmixt), aredependent on the slip between the air and the air-water mixturevelocities. If the velocities were equal, i.e., the air does notmove faster inside the pipe than the mixture, all of the rates areproportional to the correspondent inlet volumes in the pipe,fg ¼ Qg=ðQf þQgÞ, ff ¼ Qf=ðQf þQgÞ ¼ 1 − fg, and ρmixt ¼ðρfQf þ ρgQgÞ=ðQf þQgÞ, or ρmixt ¼ ρfff if ρg is neglected inthe presence of ρf . Because the air rises faster, it occupies a lower(virtual) average section inside the pipe, the air fraction is lower,and the fluid fraction and density are then superior. According toEq. (1) and the continuity equation of the steady flow, for examplebetween two successive slugs as developed in Griffith and Wallis(1961), the following expression is obtained:

fg ¼Qg

CðQf þQgÞ þ AUgbð2Þ

and thus

ff ¼ 1 − Qg

CðQf þQgÞ þ AUgbð3Þ

or

ff ¼ CQf þ ðC − 1ÞQg þ AUgb

CðQf þQgÞ þ AUgbð4Þ

Given that

ρmixt ¼ ρfff þ ρgfg ð5Þand neglecting ρg in the presence of ρf results in

ρmixt ¼ ρfCQf þ ðC − 1ÞQg þ AUgb

CðQf þQgÞ þ AUgbð6Þ

For an inclined pipe of a siphon rising leg and/or otherflow patterns, the values of C and Ugb may have to be adjusted.Woldesemayat and Ghajar (2007) performed an extensive compari-son of 68 void fraction correlations. They grouped the correlations

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in four classes in the same manner as proposed by Vijayan et al.(2000), for more than 2,800 experimental data in different flowpatterns for vertical, inclined, and horizontal pipes. They confirmedthat the methods based on the drift flux model that have a generalform analogous to Eq. (2) achieved the best results.

The two-phase flow resistance law and the corresponding fric-tion losses, even if important in an air-lift pump, are not typically ascrucial as they are in an inverted siphon, particularly for moderateair flow rates, because the length of the pumps is normally reducedand the water velocities are frequently low. The existing frictionlaws, as well as the corresponding methods (described next) arebased on experimental correlations and sometimes may incorporatesubstantive differences.

Akagawa (1957) stated that the ratio between the head loss perunit pipe length of the air-water mixture (Jmixt), referenced to waterin a two-phase flow, and the head loss per unit pipe length when thefluid alone flows (Jf), is proportional to a power (−Z) of the fluidfraction:

Jmixt

Jf¼

�1

ff

�z

ð7Þ

and determined experimentally, for ff between 0.3 to 0.5 and1.0; values for Z of 1.40, 1.90, 1.74, and 1.51, for horizontal pipesand pipes inclined to 30°, 60°, and vertical, respectively. For(vertical) air-lift pumps, Kato et al. (1975) and Todoroki andSato (1973) used the Akagawa method with Z values of 1.51and 1.75, respectively.

Another approach, Nicklin (1963) and Reinemann et al. (1990),considers that the two-phase flow follows the same resistance lawas the fluid single-phase turbulent flow. The friction factor or headloss is calculated by considering the entire mixture in the pipe aswater, which is then affected by the fluid fraction given by Eq. (4).Griffith and Wallis (1961) considered this approach with ff givenby the volumetric ratio as if no slip existed. Clark and Dabolt(1986), based on the method of Lockhart and Martinelli (1949),proposed an expedient approach, for ff ≥ 50% and slug flow,given by Jmixt=Jf ¼ 2.5 − 1.5ff. According to those authors, thissimplified equation does not introduce significant differences toeither the Lockhart and Martinelli correlation or the Nicklinapproach.

Yoshinaga and Sato (1996) used a correlation for three-phase(air-water-solids) flow similar to the equation for two-phase flowof Chisholm and Laird (1958), which can be expressed as:

Jmixt ¼ Jf þ 21ffiffiffiffiffiffiffiffiffiJfJg

p þ Jg ð8Þ

where Jf and Jg = head losses per unit pipe length of the waterand air as single phases (i.e., for the corresponding average veloc-ities Qf=A and Qg=A). In both cases, Yoshinaga and Sato (1996)used the well-known Blasius empirical equation. Jf and Jg (asJmixt) in Eq. (8) must be referenced to the same fluid.

The developed model for the two-phase flow in the siphon isindependent of the flow-resistance law that is adopted. All the re-sistance laws described, and subsequent procedures mentioned,were used on the developing and on the testing of the model.

Proposed Model Approach

Fig. 1(a) presents the inverted siphon schema and some relevantvariables before air injection. The siphon is initially dischargingwater at a steady flow rate (Qf0) between the inlet chamber(denoted as I) and the exit or outlet chamber (denoted as E) witha head loss (ΔH0). A reference horizontal plan (RHP) crosses the

section of the air injection point (denoted as A) at the pipe axis. Theinitial hydraulic head in I is given byHd0 and the water level in E byhr. The pipe has a total length L that is the sum of the descendingand rising legs of lengths Ld and Lr, respectively, with diametersDd and Dr, and cross-sectional areas Ad and Ar, that may be equal(i.e., Dd ¼ Dr ¼ D and Ad ¼ Ar ¼ A) or not.

In Fig. 1(b), an air flow rate (QgE), referenced to the local atmos-pheric pressure and temperature at the outlet, is injected at point A.The water inflow at chamber I, Qf, is maintained, Qf ¼ Qf0.When the new steady regime is established, the hydraulic headreaches Hd that will be less than, equal to, or greater than Hd0,depending whether the air injection is favorable, equal, or worsein terms of the available hydraulic head gain.

The energy equation of the single-phase flow between I and A,if ffA is the fluid fraction at A, Ld1 is equal (or approximatelyequal) to Ld, and the Coriolis coefficient is approximately equalto 1 for a water turbulent flow, yields:

Hd ¼prelA

γfþ Q2

f

2gA2r

1

ffAþ JfLd þ KI

Q2f

2gA2d

þ KA

Q2f

2gA2r

1

ffAð9Þ

Inlet chamber Exit chamberI E

Energy Line ∆Hd ∆H0

∆Hr

Qf0

Hd0 Dd Dr hr

Qf0

Ld Lr

Injection PointR. H. P. A

Descending Leg Rising Leg

(a)

I E

∆Hr ∆H0

Open channel Energy Line flow Qf0+QgE

∆Hd

Pipe flow Qf0

∆Hr hr

Hd Dd Qf0+Qg

Ld1 Dr

Lr

R. H. P. A

QgA

(b)Energy line relative to the liquid densityEnergy line relative to the mixture average density

Fig. 1. Inverted siphon schematic and energy line before and after airinjection, maintaining the water flow rate and ignoring minor losses:(a) definition of variables and initial energy line; (b) real energy lineschematic approximation after air injection while maintaining the waterflow rate (for preserving favorable measuring conditions in the experi-ments, Hd0 is raised and the descending leg always operates at the pipeflow condition Ld1 ¼ Ld)

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where prelA = relative pressure at the injection point; γf = specificweight of water (i.e., γf ¼ ρf g); Jf = fluid head loss per unitpipe length; and KI and KA = minor loss coefficients at points Iand A, respectively. In the frequent case of a hydrodynamicentrance, KI may be ignored, and depending on the mode asthe air is injected, KA may eventually be negative, meaning that anenergy gain in the injection point is possible. Due to air compress-ibility, the hydraulic head in any section of the rising leg contin-uously refers to an average mixture of specific weight different.All head calculations and numerical values require the conversionto the same density or referential. The sum ðprelA=γfÞ þðQ2

f=2gA2rÞð1=ffAÞ represents the hydraulic head in A,HA, relative

to the liquid density; see the corresponding energy line in Fig. 1(b).A (nonconventional) energy line, relative to the mixture averagedensity below each section (in the limit, i.e., between the injectionpoint and the section), is also schematically represented in thefigure. Depending on the air and water inflows and the injectionmode, the kinetic energy at section A may vary substantially.As a first approach of the implemented model, it is assumed thatffA in Eq. (9) may be set directly proportional to the entered flowrates, or alternatively determined according to Eq. (4). The absolutepressure head relative to the liquid density at the injection point(pA=γf), if the local atmospheric pressure is patm, is then given by:

pA

γf¼ Hd þ

patm

γf− JfLd − KI

Q2f

2gA2d

− Q2f

2gA2r

1

ffAð1þ KAÞ ð10Þ

Establishing the process for the gas equation, in this model anisothermal process, p=ρg is a constant, or p Qg is a constant, wherep and ρg are, respectively, the absolute pressure and density of thegas in any section of the rising leg, and because the injected airflow rate at exit conditions is fixed, the air flow rate (Qg) in anysection is given by

Qg ¼QgEpatm

pð11Þ

For a fixed geometry and Qf ¼ Qf0, since the ffA may be de-termined through QgA, and this from pA by Eq. (11), the unknownvariables in Eqs. (9) and (10) are simply Hd or pA.

The frictional head loss in the steady single-phase flow of thesiphon descending leg can be determined by any currently knownflow-resistance law for turbulent pipe flow. In the scenario of freshwater, the Colebrook-White equation usually provides the best ac-curacy, but for smooth pipes and Reynolds numbers until 100,000,the Blasius equation is also a good approach.

The energy equation in the rising leg between A and the exit Eis far more complex. Even if the process equation of the gas andtwo-phase flow resistance law (discussed in the previous section)can be established, because air is compressible, the average air flowrate in any section, average flow rate of the air-water mixture, aver-age kinetic energy, and total average weight (divided by Aγf) of thevertical column (with transversal area A) below (corresponding tothe position potential energy) vary along the pipe and are dependenton the remaining unknown energy term, which is the average ab-solute pressure that is reached in the section. Given that the absolutepressure (p) at any section is dependent on the head loss, and be-cause the flow rates vary continuously along the pipe, a differentialenergy equation with a numeric integration for the general case isrequired because an algebraic or formal integration does not seempossible.

For steady-state flow, the Bernoulli differential equation may bewritten in generic form as

dHds

¼ −J ð12Þ

where s = distance along the axis of the rising pipe. Given that theknown absolute pressure is at the outlet, the equation may be solvedby finite difference in the reverse order of the two-phase flow(from E to A):

Hiþ1 −Hi ¼ ΔxiJmixt i þ Jmixt iþ1

2ð13Þ

for i ¼ 1 to m, where m = number of distance intervals of lengthΔxi, relative to a distance x that has an opposite signal of s (and J).

The average density of the mixture, in the time and in the crosssection, varies along the pipe axis. Developing the hydraulic headterms of Eq. (13) referred to water density and considering Corioliscoefficients (defined as for single phase flows) approximately 1:�Xm−1

j¼i

�Δzjþ1

ffjþ1þffjþ2

2

�þpiþ1

γfþ Q2

f

2gA2r

1

ffiþ1

�

−�Xm

j¼i

�Δzj

ffjþffjþ1

2

�þpi

γfþ Q2

f

2gA2r

1

ffi

�

¼ΔxiJmixt iþJmixt iþ1

2ð14Þ

for i ¼ 1 to m, where Δzj and Δzjþ1 = vertical distance intervalsthat correspond with Δxj and Δxjþ1 for j and jþ 1, respectively.

The sumsP

mj¼i ðΔzj

ffjþffjþ1

2Þ and

Pm−1j¼i ðΔzjþ1

ffjþ1þffjþ2

2Þ,

with (m − iþ 1) and (m − i) parcels, respectively, represent the to-tal average weight per unit transversal area divided by γf (elevationhead) of the vertical column of the air-water mixture in therising leg below points i and iþ 1. The fluid fraction in any sectionis calculated using Eq. (4). The kinetic energy of the two-phaseflow varies significantly along the cross section and instantane-ously in time. Even ignoring the kinetic energy of the air, its aver-age value is expected to be slightly greater than ðQ2

f=2gA2rÞð1=ffÞ

when ff is obtained with Eq. (4). In the examples of the modelimplementation presented in this paper, the average kinetic energyin the cross section was also alternatively slightly increased throughthe assumption of a homogeneous mixture in the rising leg, i.e., asif no slippage occurred between the phases.

Solving to obtain (piþ1=γf), Eq. (14) yields, for i ¼ 1 to m:

piþ1

γf¼ pi

γfþXmj¼i

�Δzj

ffj þ ffjþ1

2

�−Xm−1

j¼i

�Δzjþ1

ffjþ1 þ ffjþ2

2

�

þ Q2f

2gA2r

�1

ffi− 1

ffiþ1

�þΔxi

Jmixt i þ Jmixt iþ1

2ð15Þ

or

piþ1

γf¼ pi

γfþΔzi

ffi þ ffiþ1

2þ Q2

f

2gA2r

�1

ffi− 1

ffiþ1

�

þΔxiJmixt i þ Jmixt iþ1

2ð16Þ

If the slope of the rising leg at interval i makes an angle αi withthe horizontal, then Δzi ¼ Δxi sin αi. If the Δxi space intervalsare equally spaced, and αi is considered to be approximatelyconstant, then Δz ¼ Δx sin α.

Starting from the known flow rates and pressure at the exitchamber, Eq. (16) can be solved step by step for the section iþ 1on the basis of the knowledge of all the flow characteristics at sec-tion i. However, substituting Eqs. (4) and (11), and even consid-ering a simpler form for the two-phase flow resistance law, asgiven by Eq. (7), the resulting expression is not explicit with respectto (piþ1=γf) when all flow characteristics at section i are known.

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Thus, an iterative process inside the integration step with conver-gence general requirements is required.

The iterative process may also be executed externally to the nu-meric integration of the energy differential equation. If an assay (ortrial solution) for the air flow rate diagram in the rising leg is fixed,Eq. (15) or (16) allows direct calculation of the absolute pressure inthe consecutive sections of the rising pipe and the consequent pres-sure diagram, starting from each pressure of the previous section.A new diagram of air flow may now be obtained using the processequation. An iterative process using a successive approximations(approaches) method can then be performed.

At the first iteration, the pressure diagram is linear between theatmospheric pressure at the outlet and the absolute pressure at in-jection point (pA) that is set equal to the absolute pressure immedi-ately upstream of A, forHd ¼ Hd0. The first iteration for pA is thengiven by Eq. (10) with Hd ¼ Hd0, Qf ¼ Qf0, KA ¼ 0, andffA ¼ 1. Eq. (11) of isothermal conditions is then applied for eachpipe section, and an initial air flow rate diagram is obtained. Theaverage fluid fraction of the air-water mixture in the rising legbelow each section may now be eventually computed and recorded.A new pressure diagram is obtained by numerical integrationthrough Eq. (15) or (16), starting from the atmospheric pressureat the outlet, and using Eq. (4). A new air flow diagram is obtainedwith Eq. (11), and the process is repeated until no change occurs inthe diagrams of pressure and air flow for the required accuracy.

This iterative process is very efficient, with very simple anddirect computations and extremely rapid convergence. After theenergy equilibrium in the rising leg have been established, and thus(pA=γf) and HA have been determined for a fixed water flow rateand air flow rate at the outlet, Hd may be obtained from Eq. (9)explicitly (if Ld1 ¼ Ld, or assuming Ld1 ≈ Ld) or implicitly, ac-cording to the profile of the descending leg (the opposite case),and subsequently an approximation may be obtained for the gainof the hydraulic head with air injection, Hd0 −Hd.

If the geometry and Hd0 was fixed for the injected air flowrate, then a change would occur in the water flow rate that wouldeither increase or decrease, while the air injection was favorable[Fig. 2(a)] or not favorable. A limit for the new water flow rateQflim is obtained by solving Eq. (9) with respect to Qf , with Hd ¼Hd0 and the obtained pA=γf . For this specific computation, the cur-rent implementation uses the Colebrook-White equation solvedwith respect to Jf, with a method of successive approximationsassociated with a classic bisection method for the root determina-tion of Eq. (9). Thus, the solution lies between Qf0 and Qflim.Because the head loss per unit pipe length of the two-phase flowis in general significantly larger than the head loss per unit pipelength that occurs in the descending leg (when Dd ¼ Dr ¼ D0),the solution is potentially much closer from Qf0 than Qflim.

Because of the extremely rapid convergence of the previouslydescribed iterative numerical integration, a new external iterativecycle was developed and implemented for water flow rate determi-nation. A method that can be conceptually classified as an associ-ation between the successive approximations, and the bisectionwith a balanced factor (methods) was devised and implemented.

Starting from an initial solution Qf0, the equilibrium diagramsof air flow rates and absolute pressures (including pA=γf) are cal-culated in the rising leg by the previously described model. Awaterflow rate limit is calculated with the energy equation between theinlet chamber and air injection point, with the obtained (pA=γf),presuming that the new water flow rate in the descending leg doesnot interfere with the equilibrium that was reached in the rising leg.A new water flow rate, given by Qf ¼ ðQf0bþQflimÞ=ðbþ 1Þ,where b is an integer between 1 and 20, is used as the new Qf0

for the next iteration (if b is set equal to 1, the new Qf0 lies inthe middle of the interval). The integration of the energy differentialequation and the new equilibrium of energy for the rising leg areobtained, a newQflim, narrower root interval, andQf are calculated,and the process is repeated until no more changes (in the successivewater flow rates) are observed.

The air injection benefits in terms of invert siphon design diam-eter were also analyzed. Fig. 2(b) presents a simplified energy line.The process is identical that previously described for the water flowrate. Hd0 and Qf0 are now fixed, and instead of Qf, successivediameters D, assumed to be equal for both legs of the siphon,D ¼ Dd ¼ Dr, may be computed in the iterative process. Begin-ning with the initial geometry (D ¼ D0) and the subsequent equi-librium reached on the rising leg, Eq. (9) is solved now with respectto the diameter, giving a Dlim that is used with a balanced factor forthe new iteration. The integration method of the rising leg is thenapplied again for the new diameter D obtained, giving the requiredequilibrium in section A (and the new pA=γf and HA reached).A new Dlim, narrower root interval, and D are calculated, andthe process is repeated until no more practical changes in the diam-eter (and in pA=γf) are obtained.

I E Energy Line

∆Hr ∆Hd ∆H0

Qf+QgE

Qf

∆Hr

Hd = Hd0 hr

Dd=Dr=D0

Qf+Qg

R. H. P. A

QgA

(a)

I E Energy Line

∆Hr ∆Hd ∆H0

Qf0+QgE

Qf0

∆Hr

Hd = Hd0 hr

D=Dd=Dr

Qf0+Qg

R. H. P. A

QgA

(b)Energy line relative to the liquid densityEnergy line relative to the mixture average density

Fig. 2. Inverted siphon energy line after air injection, maintaining Hd0

and ignoring minor losses: (a) energy line schematic approximation,assuming Dd ¼ Dr ¼ D, maintaining the diameter (D ¼ D0), andincreasing Qf; (b) energy line schematic approximation, assumingDd ¼ Dr ¼ D, maintaining Qf0, and reducing the diameter of thesiphon barrel (D < D0)

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Model Implementation

The numerical model described in this paper was implemented andtested on small-capacity computers. Although several alternativeswere used, the results described and analyzed in the next sectionswere obtained by considering the following: (1) a two-phase iso-thermal flow dominated by the water temperature in the testedpipes, (2) values for C and Ugb in the equations, given by 1.2 and0.35

ffiffiffiffiffiffigD

p, respectively, and (3) the flow-resistance law of the

Chisholm and Laird (1958) type expressed with Eq. (8), with Jfand Jg computed for the water and air average velocities in eachsection as single phases, using the Colebrook-White and Blasiusequations for turbulent flows, respectively.

An additional tolerance for the solubility of air in water was alsomodeled and implemented, but because some of the tested examples,presented here, are based in enclosed circuits (looped), this toler-ance was not mobilized in the examples (assumption of a residualeffect or that the water is already close to the saturation point).

The fluid fraction required for the kinetic energy computation inany pipe section of the air-water flow was considered predomi-nately as if the mixture was homogeneous.

At the current stage of the research, the model was written inFORTRAN language, with exception for the diameter decreaseprocedure that was initially tested using Microsoft Office Exceland proved to be effective.

Application of the Numerical Model to Air-LiftPumps

The model was tested for the air-lift pumps described in Yoshinagaand Sato (1996), Khalil et al. (1999), and Kassab et al. (2009), andthe results were compared with the corresponding experimentaldata presented in those papers. The results of this comparisonare summarized in Fig. 3. The water and air superficial velocities

(Qf

A and Qg

A ) are normalized in the axis of the figure through the non-dimensional parameters ðQf=AÞ=

ffiffiffiffiffiffigL

p, and ðQg=AÞ=

ffiffiffiffiffiffigL

p, where

L = vertical length of the pumps subject to the two-phase flow(equivalent to Lr and hr of an inverted siphon). A standard temper-ature of 20°C was assumed for simplicity, and the volumetric flowwas referenced to that temperature and to the standard sea levelatmospheric pressure.

The data obtained show generally a good approximation of themodel to the published experiences. The kinetic energy parcel inthe short pipes of the air-lift pumps seems to have an importantrole, principally for high ratios between the air and water flow rates.

The slight modification in the kinetic energy introduced by thedifferent fluid fraction computation introduces little change in therising branches of the curves, but comparatively important changeswhen the air rate injected increases (Fig. 3).

Experiments Performed in Inverted Siphons andComparison with the Corresponding ComputationalResults

A two-barrel siphon experimental installation was constructed andsubject to air injection in its rising branch. Fig. 4 presents thegeneral schema of the experimental apparatus. The barrels workseparately, i.e., receiving alternately all of the incoming flow inthe inlet chamber, and their sections are kept constant along thelegs. The barrels are a 35-mm internal diameter plasticized polyvi-nylchloride (PVC), commercially designated crystal tube (transpar-ent), with a fixed Ld ¼ 10.65 m and Lr ¼ 10.85 m, over a 21.5-mtotal length, and a 0.4-MPa low-density polyethylene (LDPE),nominal diameter (ND) = 110 mm, with a 94.5-mm internal diam-eter, fixed Ld ¼ 20.35 m and Lr ¼ 12.0 m, over a 32.35-m totallength. Either the 35-mm or the 110-mm pipe entrances in the inletchamber can be oriented on a straight line with the incoming flow,as diverging with an angle of approximately 45° (Fig. 4). The outletwas conceived with an extended tee accessory of enlarged sectionto allow, in any circumstance, a free discharge at atmosphericpressure.

The upstream chamber was powered by three pumps that wereinstalled in parallel through a common force main. The T accessoryin the outlet separates the incoming air and delivers the water toa first tank with two compartments separated by a stabilizing plate.The second compartment ends in a triangular weir and has agraduated rule upstream, which is regulated according to the weircrest for measuring the water level and computing the steady waterflow rate. The water is then discharged to the pumps feed tank thatencloses the circuit.

Air can be introduced inside the barrels alternatively bya 15-mm internal diameter accessory that is perpendicular to theaxis of the pipes (for the 35-mm pipe, through a tee of equal diam-eter with a reduction to the accessory), or by an 8-mm exit diameterinjector, inserted in another compatible accessory (see Fig. 4), bothfed by a plasticized PVC pipe of 8-mm internal diameter. A 2.25-kW air compressor, with a 50-L tank, feeds the air to the pipethrough a chamber that connects to a converging nozzle whichworks on sonic conditions at the throat. A pressure gauge anda thermometer connected to the chamber reported the relative

Fig. 3. Comparison of the numerical model with published experimental data from air-lift pumps

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pressure and airtemperature, respectively. Several converging noz-zles with varying diameters in the throat (1.05, 1.5, 2.0, and2.5 mm) were used between the chamber and the 8-mm PVC pipeto maintain the sonic condition, for the steady air flow rates thatwere possible to obtain with the compressor and the correspondingpressures required (and possible) in the chamber (the relative pres-sures varied between about 20 to 25 or 30 N=cm2 and 60 N=cm2).

The hydraulic head in the upstream chamber was controlled ac-cording to a differential manometer air-water connected to thechamber and to a control recipient leveled with the first for a steadywater flow in the siphon before air introduction. In the experimentsperformed (Diogo and Gomes 2011), the upstream chamber hy-draulic head and discharge elevation were kept constant for severalinitial average velocities of the water Uf0 (1.0, 0.5, and 0 m=s) andthe corresponding water flow rates for the barrel. The energy gradeline in the barrels (with the necessary adaptations) corresponds withthe situation described in Fig. 2(a). For each injected air flow rate,the water flow was adjusted through the water control valves tomaintain the initial hydraulic head in the upstream chamber.Once the equilibrium in the differential manometer was reached,

the resulting steady water flow rate was measured through the ruleon the triangular weir.

A considerable amount of the total time (15–30 min) was typ-ically required for the establishment and measurements in eachsteady state flow (equilibrium state in the manometer). After equi-librium was reached, the corresponding measurements were aver-aged for periods not less than 5–10 min. All the results presentedherein are relative to stable two-phase flows. The flow regimeswere sometimes difficult to distinguish with definitive evidencethrough the 35-mm transparent plastic pipe. Spatial variationsalong the inclined profile of the rising leg and hybrid characteristicswere frequently observed. Nevertheless, with the air flow rate in-crease it was possible to identify by trend analysis (at least parti-ally) several flow patterns, particularly patterns similar of thosethat are frequently categorized in vertical pipes as bubble (withbig bubbles), predominately slug and slug to churn, or eventuallychurn.

Qf in the triangular weir was assumed to be proportional to h2.5,where h = reading on the upstream rule on the triangular weir, and acalibration test resulted in

8Exit chamber Inlet chamber

E I

TriangularWeir 1

Rising Leg Descending Leg

Pumps Water7

Feed Tank 2

Dr Dd

Lr Ld

PVC35 mm

5 AirLDPE Air 4 6 Compressor LDPE110 mm A 110 mm

3Injection Point

To Inlet Chamber From Feed Tank

Detail Injector A Detail Inlet I

8 mm exit internal diameter

Detail Exit E1. Graduated rule2. Water flow control valves3. Converging nozzle4. Thermometer 5. Air pressure gauge 6. Air flow control valve7. Air-water differential manometer8. Water level control container

Fig. 4. General schema of the experimental setup and details of air injection, inlet chamber and outlet discharge

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Qf ¼ 1.40 h2.5 ð17Þ

Qg, for an assumed isentropic flow in the converging nozzle anda reference air density ρg, is given by (Franzini and Finnemore1997)

Qg ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikR

�2

kþ 1

�½ðkþ1Þ=ðk−1Þ�s

πD21

4

p0

ρgffiffiffiffiffiT0

p ð18Þ

where D1 = diameter of the throat; p0 and T0 = absolute pressureand absolute temperature, respectively, in the measuring chamber(stagnation point); and R and k are the gas and adiabatic constants.

The relative propagated errors (ΔQf=Qf andΔQg=Qg), assum-ing no uncertainty in the constants, can be determined by:

ΔQf

Qf¼

�1þΔh

h

�2.5 − 1 ð19Þ

and

ΔQg

Qg¼

�1þΔD1

D1

�2�1þΔp0

p0

���1þΔT0

T0

�0.5 − 1 ð20Þ

where (Δh=h), (ΔD1=D1), (Δp0=p0), and (ΔT0=T0) = relativeerrors in h, D1, p0, and T0, respectively, that were estimated inDiogo and Gomes (2011), with maximums of 1.4, 2, 4, and0.5%, respectively. According to these estimates and Eqs. (19)and (20), the maximum propagated errors in Qf and Qg are thengiven by 3.5 and 8.5%, respectively.

Table 1 shows the experimental setup working conditions, pre-viously to the air injection, for the different initial water average

Fig. 5. Comparison of the numerical model with experimental data measured in a 35-mm barrel inverted siphon

Table 1. Experimental Conditions Tested: Rising Leg Height, Water Temperature and Upstream Head

Siphon barrelRising leg height(pipe axis) hr (m)

Water initial velocity

Uf0 ¼ 0 m=s Uf0 ¼ 1 m=s Uf0 ¼ 0.5 m=s

Water temperatureT (°C)

Upstream headHd (m)

Water temperatureT (°C)

Upstream headHd (m)

Water temperatureT (°C)

Upstream headHd (m)

35-mm PVC 3.333 26 3.316 25 4.103 21 3.553110-mm LDPE 3.540 30 3.493 30.5 3.967 29.5 3.672

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velocities tested (hr is referenced to the pipe axis at the dischargesection). The 8-mm injector was used by default. The superficialwater velocity obtained in the experiments is plotted in Figs. 5and 6 against the superficial air velocity, with Qg referenced tothe standard sea level atmospheric pressure and the standard tem-perature of 20°C (i.e., ρg ¼ 1.205 kg=m3). Vertical and horizontalbars in Figs. 5 and 6 are the maximum deviations in accordancewith the error analyses described previously.

The numerical model was applied to both barrels of the siphonfor the entire range of measured air flow rates, and the resultingcurves are also represented in Figs. 5 and 6 for comparison. A null

roughness was considered for the 35-mm transparent pipe, and aroughness of approximately 0.040 mm, previously obtained forthe 110-mm LDPE when it was fully clean, was initially assumed.No relevant minor losses were presumed to exist for both pipes,with exception for a tee air entrance in the 35-mm barrel, in whicha kinetic coefficient (KA) of 0.5 was assumed and the piezometriclevels at both discharges of the pipes were initially set at their axis.

For the set of parameters used, the water flow rates obtainedby the model are very close to the experimental measurements forthe 35-mm pipe (Fig. 5) and slightly overestimated for the 110-mmpipe (upper curves of Fig 6).

Fig. 6. Comparison of the numerical model with experimental data measured in a 110-mm barrel inverted siphon

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As a first hypothesis, the differences observed for the biggersection could be explained by a different position of the null relativepressure point at the discharge and by an increase of the pipe rough-ness that seemed to happen in this siphon barrel, since it was fullof water for a long period without flow (see the discrepancy whenthe air flow is null). A maximum roughness of 0.160 mm and thepiezometric level at the top of the conduit were considered as themaximum tolerances for the deviations, and the correspondingcurves were plotted (lower curves of Fig. 6). Because the dischargepiezometric level seems to decrease when the initial flow rate in-creases, and the true pipe roughness when the experiments wereperformed is not absolutely known, the results, though a little moreproximate, do not seem to be fully conclusive.

A supposition that the differences perhaps could be explainedby eventual minor losses, particularly in the inlet chamber andinjection section (with the 8-mm injector), was also thoroughlyinvestigated. However, the results did not support that possibility,especially because the pipes were aligned in the chamber and thewater average velocity and kinetic energy were relatively small forthe 110-mm pipe. In contrast, experiments for obtaining the gain ofhydraulic head for fixed initial water flows showed clearly that thetwo forms as the air was introduced did not produce significantdifferences (Diogo and Apóstolo 2012). The 8-mm injector didnot always improve the behavior relative to the perpendicular en-trance (or in T). In some circumstances, particularly for the lowerwater flows in the pipes, the efficiency may even slightly decreasebecause of an eventual local increase of the slippage between thephases.

These small discrepancies suggest that principally for largersections, further research must be performed, particularly with re-spect to the resistance law of the two-phase flow, fluid fraction inthe rising branch, and influence of the siphon geometry. As a firstapproach, it is likely that a loss of efficiency may occur when thepipe diameter increases and the geometry is less favorable for theair-water mixture with the creation of air corridors, particularlyfor smaller hr=D quotients and/or the predominance of bubbleflows. This may explain eventually the fact that both sets ofparameters used in the model do not bound globally very well theexperimental data for the 110 mm pipe, because the hr

D ratio was just37.5, against 95 for the 35 mm pipe, and much bigger than 50 forthe air-lift examples, and perhaps due to the likely prevalence ofbubble flows in the opaque pipe, but no substantive researchwas performed yet in inverted siphons to support completely thispossibility.

The data plotted in Fig. 5 suggest a rising curve followed by adescending curve, indicating that an optimal air flow rate may exist,which carrying a maximum water flow rate for each initial averagevelocity and fixed hydraulic head. These experimental and numeri-cal results also indicate that the water rate relative variation in-creases substantively when the initial average velocity of thewater decreases. Because the air superficial velocity tested is lesserfor the 110-mm pipe, all the points in Fig. 6 are still in the risingbranch. Similar experimental curves were obtained for the gain ofhydraulic head with air injection for fixed initial water flows Diogoand Apóstolo (2012). Analogously, the relative head gains increasesubstantively when the initial water flow decreases. Thus, the hy-draulic performance of a sanitary inverted siphon with air injectionis substantially improved exactly when the wastewater flows arereduced and the self-cleaning velocities and aeration requirementsare more critical. For a typical siphon, the hydraulic gradient is keptpositive, and there is no need to insert air continuously, but onlyfor the lower water flows. This is an important issue becausethe efficiencies of the air-lift pumps are generally low, and energyconsumption would become a priority concern.

Based on a conventional design of inverted siphons and self-cleaning velocities, and assuming that the wastewater elevationis not a priority goal, an average air flow rate of the same magni-tude of the initial average water flow (for the beginning of thesiphon exploration) was advanced as a basic requirement foran air injection solution in the framework of Diogo (2008). Theexperimental and numerical results obtained in the sequenceof the developed studies seem to conceptually confirm this firstgeneric approach.

If the air flow rate to be inserted was ruled only by the hydraulicand energetic performance of the siphon, the optimal air flowcould be determined through the overall efficiency of the process(eventually with the aid the present numerical model) for a givengeometry and water inflow rate, similar to the efficiency of anyconventional pump. However, the minimization of the costs of in-vestment, energy and operation, the requirements of self-cleaningvelocities and aeration, associated with particular geometries andfrequent variation of wastewater flows, turns the selection of theair flow rate to be inserted, and its variation in time, into a nontrivialoptimization problem.

At the current stage of the research, no attempt for applying thesimilitude laws between the laboratory siphon and a large-scaleinfrastructure was performed, since it is well-known that sometwo-phase flow aspects that would occur in larger geometries can-not be well represented on a smaller scale. Also, there is no geo-metric similarity between the projected siphon in Santiago Islandand the laboratory setup, in which the diameters, lengths, heights,remaining geometry, and measurement devices were limited pre-dominantly by economic concerns. The parameters used in testingthe model, particularly the values of C,Ugb, and the flow resistancelaw (the model is independent of the values established) may needto be adjusted to better reproduce the nature and features of a large-scale two-phase flow, if a more precise computation is required.The inertial and gravity forces at a macro-scale, for the invertedsiphon pipes of 35 mm and 110 mm, seem to be generically pre-dominant in relation to the surface tension forces of the liquidphase. The dependence of the measurements and main results ob-served relatively to the corresponding non-dimensional numbers ofWeber or Eötvös seems to be restricted.

Conclusions

A model based on the numerical integration of the energy differ-ential equation for a steady-state two-phase flow was presentedin this paper. The results of the application of the model showeda good approach to several experimental data available in the liter-ature for air-lift pumps. Often, the influence of the flow resistancelaw for this equipment is relatively limited because of the frequentlysmall length of the pumps. On the other hand, the two-phase flowkinetic energy prediction seems to be somehow relevant, particu-larly for high rates of air mass introduced in the vertical pipe.

The experiments performed in an inverted siphon for a constanthydraulic head available in the inlet chamber and a fixed outletlevel show a substantive increase in the water flow rate and averagevelocity in the pipes when air is injected in the basis of the risingbranch. The results seem to confirm the potential of the techniquewhen applied to such wastewater infrastructure. The numericalmodel presented in this paper, for the set of parameters used, givesa good agreement with the experimental data measured for a35-mm plasticized PVC smooth pipe. However, very slight discrep-ancies were introduced for the 110-mm LDPE pipe. The differencescould not be fully explained with the observed increase of pipe ac-tual roughness and with the position variation of the null pressure

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point at discharge, but potentially with the barrel geometry and par-ticularly the smaller ratio between the height of the rising branchand internal diameter.

In the tested examples, use of the correlation of Chisholm andLaird (1958) typically yielded the best results, particularly for thelarger quotients between the average superficial velocities of air andwater. However, for air flow rates that were small or moderate in therising branch of the curves, use of the other resistance laws andprocedures presented in this paper did not generally yield substan-tial differences.

The model is simple to use and is independent of the set ofparameters selected, which may need to be adjusted for betterrepresentation of particular geometries. Further research, particu-larly concerning the slip between the air and water average veloc-ities and the two-phase flow resistance law (open research fields),are necessary, mainly for siphons with pipes of bigger sectionsand small ratios between the height and diameter of the risingbranch.

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