Modeling-of-Water-Pipeline-Filling-Events-Accounting-for-Air-Phase-Interactions

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/263965679Modeling of Water Pipeline Filling Events Accounting for Air Phase

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/263965679

Modeling of Water Pipeline Filling Events Accounting for Air Phase Interactions

Article  in   Journal of Hydraulic Engineering · September 2013

DOI: 10.1061/(ASCE)HY.1943-7900.0000757

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Bernardo Trindade

Cornell University

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Jose G Vasconcelos

Auburn University

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Modeling of Water Pipeline Filling Events Accounting for

Air Phase Interactions

Bernardo C Trindade1 and Jose G Vasconcelos, A.M.ASCE2

Abstract: In order to avoid operational issues related to entrapped air in water transmission mains, water refilling procedures are often performed carefully to ensure no pockets remain in the conduits Numerical models may be a useful tool to simulate filling events and assess whether air pockets are adequately ventilated However, this flow simulation is not straightforward mainly because of the transition between free surface and pressurized flow regimes and the air pressurization that develops during the filling event This paper presents a numerical and experimental investigation on the filling of water mains considering air pressurization aiming toward the development of a mode-ling framework Two modemode-ling alternatives to simulate the air phase were implemented, either assuming uniform air pressure in the air pocket or applying the Euler equations for discretized air phase calculations Results compare fairly well to experimental data collected during this investigation and to an actual pipeline filling event DOI:10.1061/(ASCE)HY.1943-7900.0000757 © 2013 American Society

of Civil Engineers

CE Database subject headings: Water pipelines; Numerical models

Author keywords: Water pipelines; Pipeline filling; Flow regime transition; Air pressurization; Numerical modeling

Introduction

Transmission mains are important components of water distribution

systems and a relevant concern is the safety of operational

proce-dures performed on those Among the operational proceproce-dures one

includes what is referred to as pipeline priming, the refilling

oper-ations that often follow maintenance tasks that require total or

partial emptying of the conduits During refilling procedures, the

air phase initially present in the pipeline may become entrapped

between masses of water in the form of air pockets Entrapped

pockets may lead to pressure surges in the system and loss of

con-veyance when not properly expelled through air valves However,

as will be shown, there is limited investigation on the development

of numerical models to simulate pipeline priming, particularly

involving the effects of entrapped air

Similar to other applications that involve the transition between

pressurized and free-surface flows (also referred to as mixed flows),

there are certain characteristics on water pipeline filling events that

pose challenges to the development of numerical models:

• Pipeline filling events are characterized by the transition

be-tween free-surface and pressurized flow regimes, and while

there are different approaches to simulate such transitions,

cur-rent models are limited by difficulties in properly incorporating

the interaction between flow features (e.g., bores and depression waves) or because of issues such as postshock oscillations;

• Pipeline filling is a two-phase, air-water flow problem, and models handling the two separate phases need to be appropri-ately linked The handling of the interface between air and water

is particularly challenging;

• Due to the formation of bores and the large discrepancy in the celerity magnitudes between different portions and phases of the flow, nonlinear numerical schemes should be used if bores are anticipated so that diffusion and oscillations at bores and shocks are minimized;

• At certain regions of the flow, particularly in the vicinity of curved air-water interfaces, shallow water assumptions are not applicable due to strong vertical acceleration because the pro-blem is intrinsically three-dimensional (Benjamin 1968);

• Several different mechanisms may result in the entrapment of air pockets during filling events, yet because conditions leading to such entrapments are still not fully understood, these cannot be properly implemented in numerical models

Related studies on the interference between air and water in closed conduits started as early as Kalinske and Bliss (1943), focusing on steady flows in which a hydraulic jump filled the con-duit causing air entrainment through the jump The work presented

an expression relating the amount of air entrained by the jump in terms of the Froude number of the free-surface portion of the flow upstream of the jump Recent experimental investigation on air-water interactions in air-water pipelines has led to advances on the understanding of the removal of air pockets by dragging, leading

to expressions for the required water flow and velocity that will result in the removal of air bubbles and pockets from pipes Among such works one includes Little et al (2008), Pothof and Clemens (2010), and Pozos et al (2010) The removal of these pockets, how-ever, would occur during the operation of these pipelines, and thus

is different from the air ventilation process that takes place during pipeline priming

Numerical simulation of the filling of water pipelines required the use of flow regime transition models because conduits start

1 Hydraulics and Hydrology Engineer, Bechtel Corporation, 3300 Post

Oak Blvd., Houston, TX 77056; formerly, Graduate Student, Dept of Civil

Engineering, Auburn Univ., 238 Harbert Engineering Center, Auburn,

AL 36849 E-mail: btrindad@bechtel.com

2 Assistant Professor, Dept of Civil Engineering, Auburn Univ., 238

Harbert Engineering Center, Auburn, AL 36849 (corresponding author).

E-mail: jvasconcelos@auburn.edu

Note This manuscript was submitted on December 14, 2011; approved

on March 15, 2013; published online on March 18, 2013 Discussion period

open until February 1, 2014; separate discussions must be submitted for

individual papers This paper is part of the Journal of Hydraulic

Engineer-ing, Vol 139, No 9, September 1, 2013 © ASCE, ISSN 0733-9429/2013/

9-921-934/$25.00.

JOURNAL OF HYDRAULIC ENGINEERING © ASCE / SEPTEMBER 2013 / 921

empty and will be fully pressurized by the end of the event.

Such models simulate both flow regimes, and one may classify

these models in two main types: shock-capturing and

interface-tracking models Shock-capturing models apply a single set of

equations to calculate both pressurized and free-surface flow

regimes and require the use of a conceptual model to handle

pres-surized flows using free-surface flow equations These models

are generally simpler to implement and provide seamless

represen-tation of the interaction between various flow features, but have

the drawback of generating numerical oscillations at pipe-filling

bore fronts that can be addressed by appropriate flux selection

or numerical filtering technique (Vasconcelos et al 2009a)

Interface-tracking models are more complex to implement because

they require a set of equations for each flow regime and proper

interaction between flow features is more complex to represent;

however, the resolution of pressurization bores is exact (no

diffu-sion) and there are no postshocks at pipe-filling bores

With regards to air simulation, most models used a framework

that is based on the ideal gas law, with fewer alternatives using a

discretized framework To date, most one-dimensional, unsteady

flow models that accounted for air pockets in pipelines are of the

interface-tracking type and use the lumped inertia approach to

sim-ulate the water phase, while air is simsim-ulated using implementations

of the ideal gas law Possibly the first such work was by Martin

(1976), which presented a model to compute surges caused by

the compression of air pockets at the end of an upward sloped

pipe-line Such models assume well-defined interfaces between air and a

water phases, essentially a portion of the system that would be in

pressurized water regime while the other would have the entire

cross section occupied by air In some instances of these model

applications, an orifice equation is added to the air formulation

to account for air ventilation during the compression process This

modeling approach further assumes uniform pressure head for the

entire air mass, and is subsequently referred to as the uniform air

pressure head (UAPH) model An example of other models that use

the same principles of the UAPH model includes the work

pre-sented by Zhou et al (2002), which studied the compression of

air pockets in the context of stormwater systems considering that

the air ventilation at orifices may be choked

Two other numerical modeling studies focused on the filling

water mains and have also used the interface-tracking and

lumped-inertia approaches Liou and Hunt (1996) proposed a simple

alter-native to simulate the pipeline filling characterized by flow regime

transition, but have not included effects of air pressurization The

model avoided the difficulties associated with the shock-fitting

technique by assuming a vertical interface between air and water,

implying in an instantaneous transition between dry pipe and

pres-surized flow upon inflow front arrival The second water main

fill-ing model usfill-ing interface trackfill-ing was proposed by Izquierdo et al

(1999) The model simulates the rapid startup of pipelines that are

partially filled, so that the flow admission generates the entrapment

of air pockets Air phase modeling uses the UAPH approach

with-out ventilation for air pockets Fuertes et al (2000) tested that

model against experimental data in order to validate the model with

good agreement, but the tested inflow rate is possibly too large to be

representative of pipeline priming operations

As presented, these modeling studies that combine lumped

inertia and UAPH approaches assume (1) well-defined interfaces

separating air and water phases, (2) high inflow rates as

conse-quence of the elevated driving pressure head, and (3) uniform

pres-sure in the air pockets The first assumption may be invalid in cases

in which air is not actually intruding into the pressurized flow, as

discussed in Vasconcelos and Wright (2008) The second

assump-tion usually does not hold if the filling is performed gradually

However, the latter assumption may be applicable and is addressed

in the present investigation

Chaiko and Brinckman (2002) presented a comparative study of three modeling alternatives to simulate air-water interactions in a idealized pipe-filling problem The first model solves both phases

by applying the method of characteristics (MOC) so that water characteristic lines need to be interpolated to match the grid during simulation, the second model simplifies air phase modeling by ap-plying a type of UAPH model, and the third applied UAPH for the air phase and MOC for the water phase, however calculating only the unperturbed portion of the water flow so that the characteristic lines have constant slope and match the grid without the need of interpolation The researchers run tests for a vertical setup that con-sisted of a a cylinder with an air pocket on the upper part, which is compressed by the water phase due to an increase in the water pres-sure at the bottom of the cylinder It was found that the second model alternative captured all the relevant events as well as the first model, even though the second didn’t capture small oscillations due to the reflection of the pressure wave in the air, which has

no major importance in most practical applications However, the geometry used in that study is idealized, and a question is how the obtained results are translated to a more complex setup in which initially stratified flow exists along with moving water pressuriza-tion interfaces

Two studies are presented as instances of shock-capturing models to simulate rapid filling of closed conduits In the context

of stormwater simulation, Arai and Yamamoto (2003) presented a model that performs flow regime transition calculation including a discretized air phase calculation approach The model applies the Saint-Venant equations for the water phase and the Preissmann slot to account for pressurization The model was implemented

in a simple, quasi-horizontal geometry, and air was modeled with

a set of discretized mass and momentum equations Model results compared well with experimental results and indicate the impor-tance of accounting for air phase effects during simulation The conditions for air pocket entrapment were not focused in this study, and the linear numerical scheme applied in the study (four-point Preissmann scheme) is not appropriate to adequately capture bores when Courant numbers are very low; this is an issue that is further discussed subsequently

The second study involved finite-volume models using numeri-cal schemes based on approximate Riemann solvers that overcome problems with low Courant numbers applied in the context of pipe-line filling A model based on the Saint-Venant equations was pre-sented by Vasconcelos (2007) using the two-component pressure approach (TPA) (Vasconcelos et al 2006) This model was sub-sequently used in a study that involved the filling of an actual 4.4-km-long, 350-mm-diameter water transmission main operated

by an environmental sanitation company of the federal district, Brazil (CAESB) (Vasconcelos 2007) Field measurements of inflow rates and pressure heads indicated that the model was able to capture the general trend of the filling Yet some discrepancies between the pressure measurement and predictions were attributed to the inabil-ity of the numerical model to incorporate limited ventilation condi-tions and consequently effects of air pressurization to the flow

Objectives

This paper aims to obtain further insight on air-water interactions during water pipeline filling operations, with the overarching ob-jective of developing a numerical framework that may be used to simulate a priori filling operations in pipelines and detect opera-tional issues related to the entrapment of air pockets To achieve

this objective, two numerical models were proposed that differ in

the strategy in which air is modeled Both alternatives use the

varia-tion of the TPA model presented by Vasconcelos et al (2009b) to

describe the water phase Air phase modeling is performed either

by using a discretized framework that applies the Euler equation or

by using a type of UAPH model Another objective was to assess

the benefits of using a discretized framework to simulate the

air phase

Associated with the numerical development, an experimental

program was conducted using a scale model apparatus that

incor-porates essential features of a water pipeline filling problem Key

parameters in the problem were systematically varied, including

inflow rate, pipeline slope, and ventilation degree Experimental

measurements included pressure, pressurization interface

trajecto-ries, and inflow rates Both modeling alternatives for the air phase

were compared to experimental data and to the field data of an

actual water main filling event presented by Vasconcelos (2007)

Methodology

Numerical Model

Certain flow features of the water main pipeline filling problem

were determinant in the model’s formulation so that it could

de-scribe the filling process adequately With regard to the water

phase, these features include:

• Mixed flows: Handled by applying the TPA model variation

from Vasconcelos and Wright (2009);

• Postshock oscillations at pipe-filling bore fronts: Use of a

numerical filtering scheme (Vasconcelos et al 2009a,b);

• Air pocket entrapment and pressurization: Used either the Euler

equations or UAPH model;

• Free-surface and pipe-filling bores: Use of the approximate

Riemann solver presented by Roe (Macchione and Morelli

2003); and

• Solution stationarity: Use approach presented by Sanders and

Bradford (2011)

The air phase in the model is represented by a well-defined air

pocket that is not significantly fractured This pocket shrinks due to

compression by the water phase that gradually occupies the lowest points in the pipeline profile Air is displaced and escapes through ventilation orifices located at selected locations According to Tran (2011), for such flow conditions the air compression process may

be considered isothermal and this assumption is used in both model approaches used to simulate the air phase The air phase is calcu-lated as if the only atmospheric connections occur at ventilation points, which are treated as orifices for simplicity Ideal ventilation with negligible air phase pressure head is assumed to exist prior

to the formation of an entrapped air pocket, as will be discussed subsequently When a pocket forms, it is delimited by a ventilation orifice and a flow regime transition interface, either abrupt (pipe-filling bore) or gradual In the proposed model, an air pocket is caused by the closure of a downstream valve or by the development

of a pressurization interface as water fills the lowest points of the pipeline Fig.1presents a sketch of a typical application, whereas Fig.2 presents the overall structure of the proposed model Water Phase Modeling

The TPA model, used in the water phase simulation, modifies the Saint-Venant equations, enabling them to simulate both pressurized flows and free-surface flow regimes This model has been improved

in the past years and the alternative used in this paper was presented

in Vasconcelos and Wright (2009) This alternative has a term that accounts for air phase pressure head, so the modified Saint-Venant equations are in divergence format

∂U

∂t þ

∂FðUÞ

where

U ¼

A Q



; FðUÞ ¼

Q2

A þ gAðhcþ hsÞ þ gApipehair



; SðUÞ ¼

gAðS0− SfÞ



ð2Þ

hair



¼ 0 → Free-surface flow without entrapped air pocket or pressurized flow

hs¼

8

<

:

a2 g

ðA − ApipeÞ

hc¼

8

>

>

>

>

D

33sinðθÞ−sin

3ðθÞ−3θcosðθÞ

whereθ ¼ π −arccos½ðy−D=2ÞðD=2Þ

D

ð5Þ where U ¼ ½A; Q
T is the vector of the conserved variables;

A = flow cross-sectional area; Q = flow rate;FðUÞ = vector with

the flux of conserved variables; g = acceleration of gravity; hc =

distance between the free surface and the centroid of the flow cross

section (limited to D=2); hs= surcharge head; hair= extra head due

to entrapped air pocket pressurization;θ = angle formed by free-surface flow width and the pipe centerline; D = pipeline diameter;

Apipe= cross-sectional area (0.25πD2); and a = celerity the acoustic waves in the pressurized flow

The numerical solution used in the implementation of the water phase model used the finite-volume method and the approximate Riemann solver of Roe, as presented in Macchione and Morelli (2003) This choice was motivated by the significant discrepancy

in the celerity values between the free-surface and pressurized flows, and between air and water flows This discrepancy may be

of the order of 2 or 3 orders of magnitude and yields an extremely low Courant number for the free-surface water flow Nonlinear schemes such as the Roe scheme are known to provide accurate bore predictions even in very low Courant number conditions For dry bed regions of the flow, it was assumed that the flow depth would start as a minimum water depth of 1 mm In such cases, the model would then predict the existence of a nonphysical

JOURNAL OF HYDRAULIC ENGINEERING © ASCE / SEPTEMBER 2013 / 923

flow of this thin layer down the pipeline slope To deal with this

problem, the approach presented in Sanders and Bradford (2011)

was used In this formulation, in order keep a minimum water layer

with no motion and at the same time keep the stationarity of the

solution, two criteria were followed in order to perform flow

cal-culations at interior finite-volume cells: one is based on the ratio

between friction forces and the other based on the minimum

sub-merged area of the cell After computing these criteria to all cells in

the domain, only the cells in which at least one of the two criteria is

met have the flow variables calculated Details of these

formula-tions are omitted for brevity, but may be found in the previously

mentioned paper

Two source terms were considered for the water phase

model-ing, one accounting for pipe wall friction and the other accounting

for pipe slope Both calculations followed the approach presented

in Sanders and Bradford (2011) For the pipe wall friction source

term, a semi-implicit formulation based on Manning’s equation was

used, while for the pipe slope a formulation that preserves

statio-narity of the solution was used

The upstream boundary condition for the water phase refers to

all that is inside the dashed box in Fig.1 It is based on an iterative

solution that ensures that local continuity and linear momentum at

the pipeline inlet are satisfied, regardless of the flow regime at that

location The local continuity equation for the reservoir is

dHres

where Hres= reservoir water level; Qrec= flow rate admitted into

the reservoir from the recirculation system; and Qin= flow rate that

enters the upstream end of the pipe The calculation of the updated

flow velocity at the upstream boundary cell (unþ11 ) uses an ordinary

differential equation representing the linear momentum

conserva-tion, which in turn is derived from a lumped inertia approach:

unþ11 ¼ un

1þ Δt



g Δx



Hn res− Kequn2jun

2j 2g



− ½wdepth2þ maxð0;hsn

2þ hairn

2Þ



− fun2jun

2j

un 2 2Δx

 ð7Þ

where wdepth= local water depth; n = time step index; Keq= overall

local loss coefficient in the inlet; and f = friction head loss in the

short pipe portion inside the boundary condition right after

the inlet

After the velocity in the cell is obtained, Froude number is cal-culated with the current wdepth1in order to assess if the flow is sub-critical If this is the case, wdepthnþ11 is updated according to the Hartree MOC for free-surface flows as shown in Sturm (2001)

If flow depth at the inlet is less than the pipe diameter D, then the surcharge head hs is set to zero while hair may be nonzero if there is any entrapped pocket On the other hand, if wlevelnþ11 > D, flow at inlet is pressurized, hairis set to zero, and hsis recalculated

to match the piezometric head at the upstream end, calculated with the energy equation With the depth and the head updated, the flow area Anþ11 is updated and a new flow rate is then calculated with

unþ11 and Anþ11 The downstream boundary condition used to compare the model with the experiments can be either a fully opened or closed gate valve For the case in which the valve is fully opened, the approach called transmissive condition presented in Toro (2001) is used For the case in which the downstream boundary condition is a closed valve, the boundary condition is calculated enforcing the relevant characteristic equation and zero velocity at the downstream end:

wlevelnNoþ1¼cr

where Kr = constant factor that depends on the flow conditions in the previous time step (Sturm 2001) If wlevelnþ11 > D, the flow depth at the downstream end becomes pressurized In such a case, wlevelnþ1No is set to the value of the pipe diameter, hsis set as the extra head of the cross section minus D, and hair is set to zero Air Phase Modeling

In the proposed model, air is initially considered as a continuous layer over the water layer (stratified water in free-surface flow mode) During the simulation, air is handled in one of the two fol-lowing manners:

1 At the first stage of the filling, when the air within a given pipe consists of an entire layer that is connected to atmosphere at the ventilation orifice and at its lowest point within the pipe (ideal ventilation), it is assumed that the air pressure in the entire layer is approximately atmospheric, and air velocity

is assumed negligible This condition persists until a pocket

is formed at the lowest point during the filling process

2 At the second stage, once an air pocket is formed, the connec-tion to the atmosphere at its lowest point is lost Air phase pressure is expected to develop during the filling process, re-quiring calculations with either one of the two proposed air

Fig 1 Representation of the proposed model key components

Fig 2 Flowchart for the model calculation procedures

JOURNAL OF HYDRAULIC ENGINEERING © ASCE / SEPTEMBER 2013 / 925

phase models to determine its pressure and influence in the

water flow

An algorithm was developed to track air pocket volume and start

and end nodes as it shrinks in order to simulate its behavior with

either of the two models For this, the mechanism considered for air

pocket formation is the isolation of an air mass due to the

develop-ment of a flow regime transition interface or a closed downstream

valve As mentioned, it is assumed that during a pipeline filling

event this air pocket will be delimited by a ventilation orifice

and a flow regime transition interface This interface will move

mainly toward the air pocket ventilation point, compressing the air

pocket and forcing its elimination through the ventilation orifice, as

is sketched in Fig.1

The model alternative that uses the UAPH model assumes

(1) uniform pressure in the whole air phase, (2) the validity of the

ideal gas law, and (3) isothermal air flow This model may be

expressed as

ρnVn

p¼ ρnþ1Vnþ1

air¼ Mnþ1

where Mair= mass of air within the pocket with volume Vp; andρ =

specific mass of air In order to consider the air escape or admission,

an extra term was added to Eq (9), yielding

ρnVn

p¼ ρnþ1Vnþ1

p þ Mairn þ1

where Mair out = air mass that escapes through the ventilation

orifice in that instant, calculated as presented in Eq (19) presented

subsequently

The second alternative to model the air phase uses a discretized

framework, applying a one-dimensional isothermal form of the

Euler equation:

∂U

U ¼

ρ

ρu



ρu

ρu2þ p



Sd1

Sd2þ Sf a

 ð12Þ with the pressure p defined as

whereU = vector of conserved variables; F = vector of the

con-served variables fluxes; S = vector of source terms; α = celerity

of the acoustic waves in the air; and Sd1;i, Sd2;i, and Sf a= source

terms

Applying the Lax-Friedrichs scheme (LxF) as presented in Toro

(2001) to Eq (12), one has the following expressions to update the

conserved variables:

ρnþ1

i ¼ρniþ1þ ρn

i−1

2 −2ΔxΔt ½ðρuÞniþ1− ðρuÞn

i−1
þ ΔtSd1;i ðρuÞnþ1

i ¼ðρuÞniþ1þ ðρuÞn

i−1 2

−2ΔxΔt



½ðρuÞn iþ1− ðρuÞn

i−1
uniþ1þ un

i−1

ρn iþ1− ρn i−1 2Δx



The choice for the LxF scheme was based on its simplicity and

the lack of shocks in the air phase flow

In pipeline filling problems, the mechanism causing the motion

of the air phase is the displacement of air in the cross section caused

by changes in the water flow depth underneath the air pocket This effect is accounted for in the source terms Sd, as presented in Toro (2009):

Sd1

Sd2



¼1A



∂Aair

∂t þ∂Aair∂x uair



ρ ρu



ð15Þ

where Aair¼ ðπ=4ÞD2− A and is calculated only in free-surface flow cells An explicit implementation of source terms Sd led to instability of the numerical solution, so a semi-implicit approach was applied in this paper The air phase is first calculated without considering changes in Aair, returning a preliminary solution

ˇU ¼ ½ˇρ; ˇρu
, which then needs to be adjusted with a correction fac-torϕ so that a definitive solution is achieved The definitive solu-tion and correcsolu-tion factorϕ are represented by

U ¼

ρ ρu



¼ ϕ

ˇρ ˇρu



ð16Þ with

ϕ ¼

1 þA1n i



Aairniþ1− Aairn

i

Aairnþ1iþ1 − Aairnþ1

i −1 2Δx uni −1

ð17Þ

The solution of Sd source terms presented some oscillations

at the region of the strongest free-surface flow gradients, at the vicinity of the pressurization front Two approaches were used to-gether to minimize these oscillations The first one was to limitϕ

to the range ϕ ¼ ½1.005∶0.995
The second was the application

of an oscillation filter in the air phase internal nodes, following Vasconcelos et al (2009a) withϵ ¼ 0.05 This approach resulted

in a good balance between pressure accuracy, presenting an average continuity error of less than 4% for the air phase in the comparison with the experimental cases and 1% in the comparison with the field measurements The likely source of the continuity error for the Euler equation model are the orifice boundary condition and the limitation of the correction factorϕ [Eq (16)] to a certain value, which distorts the actual required air compression due to pocket vertical shrinking However undesirable, air continuity errors did not seem to have an major impact in the overall results, as the com-parison between numerical models and experiment indicate The other source term added to the simulation of the air phase flows was the friction between the air phase and the pipe walls, as described in Arai and Yamamoto (2003):

Sf a¼faPauajuaj

where Pa = perimeter of the air flow

For the UAPH model, the boundary condition used at the upper-most point in the pipeline reach (where the ventilation valve was located) was a discharging orifice The orifice is represented by an equation similar to one presented in Zhou et al (2002):

Mairnþ1out ¼ ΔtCdAorifρ nþ1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2ρnþ1ρatm− ρatmα2

s

ð19Þ

where Cd= discharge coefficient that is assumed as Cd¼ 0.65; and

Aorif = orifice area Eq (19) was coupled with Eq (10) to yield

ρnVn

p¼ ρnþ1Vnþ1

p þ ΔtCdAorifρ nþ1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 nþ1ρatm− ρatmα2

s

ð20Þ

For the Euler equation model, two boundary conditions are required At the lower point of the pipe where the water pressuri-zation front is displacing the air, a reflexive boundary condition

(Toro 2001) was used At the uppermost point, the ventilation

or-ifice boundary condition for the Euler equation approach is similar

to the UAPH model in the sense that both apply a continuity

equa-tion along with the orifice equaequa-tion The continuity equaequa-tion for

this boundary condition is

ΔtAnþ1

air ρnþ1

1 unþ11 ¼ Anþ1

air Δxðρnþ1

1Þ þ Mairnþ1

where Mair out= air mass discharged through the ventilation orifice,

calculated using Eq (19) Eq (21) is solved for un1þ1 using the

Riemann invariants for the isothermal, one-dimensional, primitive

version of the Euler equation (Pulliam 1981):

On both models, calculations are stopped when the pocket has

evacuated 95% of its original volume When this condition is

reached, the average head of the air pocket is assigned to its cells,

remaining constant until the end of the simulation This was

mo-tivated to avert calculation instabilities caused by increasingly

smaller air pocket volumes, and follows the strategy used in Zhou

et al (2002)

Experimental Program

An experimental investigation was conducted to gather insights

on the characteristics of the pipeline filling problem with limited

ventilation, and to validate the proposed numerical model The

ap-paratus was inspired in the one presented in Trajkovic et al (1999),

yet with modifications that limited ventilation conditions

Experimental Apparatus Setup

A sketch of the experimental apparatus is presented in Fig.3 The

experimental apparatus included a clear PVC pipeline with length

L¼ 10.96 m and diameter D ¼ 101.6 mm with adjustable slope

At the upstream end, a0.66-m3 capacity water reservoir supplied

flow to the pipeline through a 50-mm ball valve; at the downstream

end flow discharged freely through a 101.6-mm knife gate valve

into a0.62-m3 reservoir, and flow was subsequently recirculated

with pumps Right after the inlet control valve, a T-junction was

installed in the pipe so that different caps with ventilation orifices

could be installed Initial steady flow conditions were such that

free-surface flows exist at the whole pipeline because the

down-stream gate was fully opened A sudden closure maneuver (within

0.3 s) of the knife gate valve at the downstream end of the pipeline

blocked the downstream ventilation, triggered a backward-moving

pressurizing interface, and resulted in the entrapment of an air

pocket These air pockets became pressurized as water accumulated

at the downstream end of the pipe pushed the air mass through the

ventilation orifice in the beginning of the pipe Two pressure

trans-ducers, Meggit-Endevco 8510B-5, were installed at the upstream

end of the pipe (x¼ x=L ¼ 1, measured from the downstream end) and at an intermediate point x¼ 0.39 Transducer results were calibrated each experimental run with the aid of four digital manometers, with of 3.5-m H2O maximum pressure head and 0.3% accuracy Flow rates were measured with a Nortek MicroADV positioned in the recirculation system and confirmed by a paddle-wheel flow meter

Experimental Procedure The experiment procedure consisted of the following steps:

1 With the desired slope set in the pipeline, the pumps were started; valves near the pump were throttled enough to provide the selected steady flow rate to the system

2 The desired ventilation orifice was installed

3 When the water level at the upstream reservoir and pipeline attained steady levels, readings were perfromed at all man-ometers, as well the upstream reservoir head

4 The data logging was started for the pressure transducers, the MicroADV, and manometer at the upstream reservoir

5 The downstream knife gate valve was rapidly maneuvered and closed, entrapping an air pocket and creating a backward-moving pressurization front

6 Digital cameras (30 frames per second) recorded the whole pipe filling process, one of them tracking the bore and another one tracking the pressurization interface

7 When the pressurization interface approached the ventilation orifice, it was rapidly closed to avoid water spilling

8 The pump was shut down and control valves closed so that pressure could attain hydrostatic conditions

9 Manometer readings were performed and data logging was stopped

The use of two cameras to track the inflow and pressurization front was particularly necessary in the cases in which interface breakdown (Vasconcelos and Wright 2005) occurred; otherwise just one camera tracking the pipe-filling bore front was used The described experimental program varied systematically the flow rates, ventilation orifice diameters, and pipeline slopes Table 1 presents the ranges of the tested experiment variables, with a total

of 36 conditions tested A minimum of two repetitions were Fig 3 Sketch of the experimental apparatus used in the investigation

Table 1 Experimental Variables Variables Tested range Normalized range Flow rate 2.53, 3.79, and 5.05 L=s 0.245, 0.368, and 0.490 Slope 0.5, 1, and 2% Not available Ventilation

orifice diameter

0.63, 0.95, 1.27, and 5.06 cm

0.0625, 0.09375, 0.125, 0.5 Note: Flow rate normalized by Q  ¼ Q= pffiffiffiffiffiffiffiffigD 5

and ventilation diameter

by d  ¼ d orif =D.

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performed for each condition to ensure consistency of the data

collected

Results and Discussion

The experimental results are compared with the proposed

num-erical model using both approaches to simulate the air phase

pressure The numerical model predictions are also compared with

the field data collected by Vasconcelos (2007) during an actual

re-filling operation of a 4.4-km-long, 350-mm-diameter, ductile iron

pipeline in Brasilia, Brazil, operated by the waterworks

com-pany CAESB

Experimental Results

Fig.4shows the pressure history close to the ventilation orifice for

all tested cases in the experimental program with normalized orifice

diameter dorif ¼ dorif=D smaller than or equal to 0.125 The trans-ducer at that station was located at the pipe crown, so it measured air phase pressures for most of the filling processes As would be anticipated, higher pressurization levels were observed for smaller ventilation orifices, while the filling time was smaller for higher flow rates

Air phase pressure head results were not significantly different for varying pipeline slopes On the other hand, there was a slight difference in the filling time between different pipeline slopes for a given ventilation orifice and flow rate This difference is attributed

to the different initial water levels in the apparatus prior to the clos-ing of the knife gate valve at the downstream end Also, as can be noticed in Fig.4, the smallest ventilation air phase pressure head kept increasing during the filling process, indicating steady flow for these cases was not attained

Fig 5 presents the pressure head hydrograph for x¼ 0.39 (measured at pipe crown) for experimental runs with dorif ≤ 0.125 There is a sudden step up in the pressure values the moment at

Fig 4 Measured air phase pressure heads for all tested conditions where dorif <0.5

which the flow regime transition interface reached that station.

As in the case of pressure measurements at the ventilation orifice,

these pressures kept increasing due to the increase in the air

pressure for the smallest ventilation case The magnitude of the

jump in the pressure readings was an indication of the strength

of the pipe-filling bore front, which increased for larger inflow rates

and ventilation orifices The absence of this discontinuity was a

sign of either gradual pressurization interface and/or the occurrence

of interface breakdown feature due to the interaction between the

backward-propagating pipe filling bore front and the depression

wave generated at the pipeline inlet by the air pressurization

The relevance of the interface breakdown feature is that its

occur-rence may pose difficulties to the application of pipeline filling

models that use well-defined inflow interfaces as a modeling

hypothesis

To illustrate the impact of the interface breakdown feature

to results, Fig 6 presents two sets of trajectories of moving

pressurization interfaces for normalized flow rates of Q¼

Q=pffiffiffiffiffiffiffiffigD5

¼ 0.245 and 0.490 and 2% slope, measured for all tested ventilation diameters All these interfaces start as pipe-filling bore fronts at x¼ 0 as the gate valve is closed Such bores lasted until

x≈ 0.28 when they encountered the depression wave originated from the upstream end of the pipe For both flow rates, an interface breakdown feature was noticed when the smallest ventilation ori-fice was used On occurrence of the feature, the pipe-filling bore became an open-channel bore that moved more slowly toward the ventilation orifice, leaving an air intrusion on its top Interestingly,

as the backward-moving bores approached the ventilation orifice, there was an acceleration on their motion, regardless of whether interface breakdown occurred or not The cause for this change

in bore velocity is not determined yet

When interface breakdown occurred, interface measurements included both the position of the open-channel bore and the pressurization front The latter was a gradual transition, and

Fig 5 Pressure head variation at the pipe crown for x¼ 0.39 for all tested conditions where d

orif<0.5

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