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7/27/2019 Popescu Stefan

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IWM 2008 CONFERENCE

177

MODELLING OF A HYDROPHORE PUMPING FACILITY INSLOW VARIABLE OPERATIONAL REGIMES

tefan POPESCU1,Nicolae MARCOIE2, Daniel TOMA3

The paper presents the conceptual and mathematical models which are

describing the functioning of a hydrophore pumping facility, working in quasi-

permanent and slow-variable transition regimes. This, achieved within the

systems theory, serves to compute the main power and economic efficiency

parameters of water pumping, and those for pumps starting and stopping

sequences.

Keywords: mathematical models, transition regimes, modeling, water

1. Introduction

One frequently used way to adapt pumps to the variable demands of watersupply networks is the intermittent operation of non-adjustable pumps and thehydrophore based compensation of flows. If such facilities are soundly designedand operated they can adequately satisfy the networks demands in terms of flowsand loads, they can provide a decrease of the pumps starting sequencesfrequency and they can ensure the networks protection against surge damages.

Below we present the modeling of a hydrophore pumping facility, inquasi-permanent and also in slow-variable transition operating regimes.

2. Conceptual model

The considered hydrophore pumping facility draws water from a suctiontank (vessel), having a free constant level, and delivers it directly towards a

pressurized water supply network. This networks flow and load demands areconstant during the analysis (study) period. The pumping basic technological line(PBTL) is equipped with 2

tipN non-adjustable pumping devices (PD), as it

follows: mpI > 0 - main PD i mp

I I 0 - auxiliary PD; the m mpI

pII+ PD can work

1 Prof.dr.ing. Faculty of Hydrotechnics, Geodesic and Environmental Engineering, Department ofLand Reclamation and Environmental Protection,Technical University Gh. Asachi of Iai2 Assoc. Prof. Faculty of Hydrotechnics, Geodesic and Environmental Engineering, Department ofLand Reclamation and Environmental Protection, Technical University Gh. Asachi of Iai3 Assist. Prof. Faculty of Hydrotechnics, Geodesic and Environmental Engineering, Department ofLand Reclamation and Environmental Protection, Technical University Gh. Asachi of Iai


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MODELLING OF A HYDROPHORE PUMPING FACILITYIN SLOW VARIABLEOPERATIONAL REGIMES

178

in parallel and are constituted in the so-called [3, 4] hydraulic generators battery(HGB). A variant of functioning for the HGB, , is described via the number of

PD of each type, in function, ( m mpI

pII

, ) , where 0 m mpI

pI and

0 m mpII

pII . The total number of HGBs operational variants,Nv (including

the variant m mpI

pII

= = 0), is N m mv = + +( ) ( )pI

pII1 1 .

The hydraulic system represented by PBTL and hydrophore,schematically, featuresNsc=3 characteristical sections (Fig. 1), namely: k=1 viaHGBs delivery collector; k=2 via the common delivery pipes connection withhydrophore R2, considered to be cylindrical, with a horizontal generator (Fig. 2);

k=3 via the delivery pipes connection that is connecting it to the served watersupply network. Depending on served water networks needs and the PDsfunctional parameters, this hydraulic system can operate in a continuous way(quasi-permanent regime) or in a discontinuous way (slow variable transitionregime).

Fig. 1. Diagram of hydraulic system PBTL-hydrophore

Considering its functions, according to [1], in a hydrophore having a totalvolume VH, the next characteristical volumes can be described (Fig. 2):

Fig. 2. Geometrical features of a horizontal cylindrical hydrophore


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A water volume needed to prevent the weakening of air cushion, Vo; Ananti-surge volume, Vp ; A volume available for flow compensation, Vu , and the

volume of compressed air, Va , required vor providing the load in the tank, 2RH .

3. Mathematical Model

The conceived mathematical model is based on the laws which govern thetransformation of compressed air volume within the hydrophore, the way in whichHGB works together with the water supply network, and, as well, the way inwhich the generating hydraulic machine works together with the electric drivingmotor, from a mechanical point of view.

In order to fully satisfy at least the water supply networks demands, thepiezometrical loads within section k= 2 , ( )tam2H , in delivery collector, HC(t), andwithin compensation tank,

( ) Rf ptzt ++=)(HR (1)

have to satisfy the following restrictions:

( ) ( ) ( ) ( ) minRav2HF;

maxRRH

minR;F

am2HCH HtHHtHHtt (2)

For the situation ( ) ( )H Hav F Fmin

3 t t H = , variants ( ) , , , ,= 1 2 Nv , are to

be ordered in an augmenting manner function of the HGBs flow, Q ; thereforethe next inequalities are satisfied:

( ) { }1,,1,Q0 v1vv (5)

( ) .pentru,H F1vminR

am2 * QQHt =< (6)

In this case we can describe the next: 10 - the compensation tanks fillingstage and 20 - the compensation tanks draining stage.


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During the filling stage, with duration Tu, HGB works in the variant *,

and theHRload increases from initial value HRmin to final valueHR

max .

During the draining stage, with duration Tg, HGB works in variant * -1,

and the piezometrical load HR is decreasing from initial value HRmax to final

valueHRmin .Obviously, for the whole cycles duration, it results:

gu TTT += (7)

Next we will present, classified in groups with remarkable technicalmeanings, firstly the computing equations needed to model the pumping facility

for a continuous regime (quasi-permanent), and next the additional and/or specificequations (corresponding to a variable slow transition regime), which occur in thecase of cyclic functioning.

3.1. Mathematical model in case of continuous functioning

This model is described by means of equations given at items a)...l).a) Pumps characteristics at variable speed [3, 4]:

- for load,

( ) ( ) ( ) { };III,,,H 2pp2

p ++= iQcnQbnanQii

Hiii

Hii

Hiii

(8)

- for power,

( ) ( ) ( ) ( ) { }N , , I, II ;p p pi i i Ni i i Ni i i Ni i Ni iQ n a Q n b Q n c n d n i= + + + 2 2 3

(9)

- for torque,

( ) ( ) ( ) { }III,,, 2p2

pp +++= indcnQbQanQiiiiiiiiiii

MMMMM (10)

b)Asynchron motors characteristics [3, 4]:

- for torque,

( )( )

{ };II,I,

1121

m ++

= i

cnbn

nnan

iiii

iis

iii

M (11)

- for power factor,

( ) ( ) { }2

m m mcos , with I, IIi i i ia b c i = + +

i i iM M M

(12)


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181

- for cost efficiency,

( )( ) ( ) ( ) iiii

i

i

iii

fnne

n

jjrjn ~~tgtg~1

1,,

m22

m

m

nem

1

++

+=

ii

ii

MM

MM (13)

c) The water networks equivalent characteristic (NEC),

(against section k= 3 ) [3]:

( ) ( )pQMHQ FechstechF3H += (14)

d) Energy equations:

- between sections k=1 and k=2,

( ) ( ) ( ;hHH 11,2rCam2 avQtt = (15)- between sections k=2 and k=3,

( ) ( ) ( ) ,QhHH av23,2

rav2

am3 ttt = (16)

Applying technique given in [4], from equations (8)(11), the nextelements have been computed:

e) Pumps characteristics , at quasi-constant speed:- for load

( ) ( ) { }III,,H 2ppp ++= iQcQbaQ iiHiiHiHii (17)- for power,

( ) ( ) { }III,,2ppp ++= iQcQbaQN iiNiiNiNii (18)- for torque,

( ) ( ) { }IIIiQcQbaQ iiiiiii ,,2ppp ++= MMMM (19)- the speeds variation laws,

( ) ( ) { }III,,n 2ppp ++= iQcQbaQ iiniininii (20)f) The reduced load characteristics (in delivery collectors section):

( ) ( ) { }III,,H 2ppp ++= iQcQbaQc iiHiiHiHii (21)

g) HGBs equivalent load characteristic:


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- for variant , which gives ( ) { } :III,,with,00 pp jijimmji

= andand

( ) 22

Q

m

cQ

m

baQnH

ip

iH

ip

iHi

H

++=, (22)

- for variant , which gives 11IIp

Ip mm and :

( ) ( ) ( ) ( ) HHHHHHH eaQaQdaQcbQ,nH +++=22 (23)

where:

( ) ( ) ;gbaba

b;c

bm

c

bma

IIIIII

HIIH

II

H

II

pIH

I

H

I

pH

22

2

1 =

+=

( ) ( ) ( )IIIHIII

H

III

H bbge;g

aad;

g

aac ==

+=

22

22

2 (24)

and in which:

( ){ } ( ) ( )22

2

4III

iH

iHi

Hi

iH

ipi

aagiI,IIicu,c

bab;

c

ma ==

= (25)

h) Energys equations between the sections of suction tank and those of

delivery collector:- for each pump,

( { }III,,H pAC += icuQHH ii (26)- for HGB,

( ).Q,nHHH AC += (27)

i) The outline conditions in section k=1: s=s1

( ) ( ){ }T I II1 p p,m m =u

(28)

j) The outline conditions in section k=2:

am2 2 0H z p g = (29)


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( )2f .

a Hp V W

z z const

=

= + (30)

k) The outline conditions in section k=3

( ) ( ) ( ) ( ) ( ) ( ) ( )am,avav am av 3 am av av3 3 3 3 3 3r3Q Q ,Q , H H h 0 .Qt t t t t t Q= = = (31)

l) The powers and efficiencies equations:- electric motors electrical power,

( ){ }I,IIi,

j,M,n'h

QNP

iii

ip

ii

e im

=

1

(32)

- pumping facilitys electrical power,

IIe

IIp

Ie

IpeSP PmPmP +=

(33)

- hydraulic power requested by pilot network,( )AFFhRP HHQgrP = (34)

- pumping facilitys efficiency (specific equation),

[ ]%P

Ph

eSP

hRPSP 100= (35)

3.2. Mathematical model for cyclic functioning

In this case, besides equations from item 3.1.[minus equations (15) and(16) as well as (29) and (30) ], we have to take into account the next equationsand conditions:

m) Energys equation between sections k=2 andR2tanks section

( ) ( ) ( )( tQtt Q3R,am2ram2R ,hHH = (36)Where head loss ( )( )h ,r 2am,R Q tQ3 is given by equation:

( )( )( )( ) ( )

( )( ) ( )


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n)The waves dynamic equation within delivery pipe and the continuityequation for compensation tankR2:

- waves dynamic equation

( )( ) ( ) ( )( )tQQzQ

t

t

A

L

gQ3R,am

2r1,2rf*

21 hh,Hd

dQ

0 = (38)

- continuity equation for tank R2

( ) ( )tQdt

dz Q

3= (39)

o) Terminal conditions for the integration of equations (38) and (39) :

- for tanks filling stage:0300 ,0 uQuu tQt == (40)

( ) fRuu zHzz,t === minmin00 0 and ( ) fRuuu zHzz,Tt === maxmax

11(41)

- for tanks draining stage:

0300gQgug tjQ,Tt == (42)

( ) fRgug zHzz,Tt ===max

max00

and ( ) fRgg zHzz,Tt ===min

min11

(43)

p) The equations for pumping facilitys energies and average efficiency:

- power consumed in tanks operational stage ,

( )=1

0

t

t

feSP

eSP dttPE (44)

where: = u - for filling stage; = g - for draining stage;- power consumed in Tcycle,

g

eSP

u

eSPeSP EEE += (45)- hydraulic energy demanded by network, in cycle T,

( )THHQgrE AFFhRP = (46)

- pumping facilitys average efficiency,


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[ ]%EEh

eSP

hRPSP 100= (47)

Air cushions volume, Va , can be assessed in function of the water volumewithin the vessel, Vw, as it follows:

a H wV V V= w o p uV V V V = + +

(48)

In the usual cases of horizontal hydrophores (Fig. 2), water volume Vw canbe computed as a function of the total volume VH and the depth of water withinthe vessel, , as it follows[1]:

( ) ( ) ( )w w H H V V V R V A R B = = + (49)

with 2HV R L= , andA=0.60303 andB=-0.10418 for [ ]0.3,1.7R .

The precise equation of function ( )wV is given in [2]:

( ) ( ) ( )2 22 2w

L DV D arctg D D

= +

(50)

The waters extreme levels within hydrophores tank, that is min andmax , have to correspond to the extreme loads within tank, that is minRH and,

respectively, maxRH ; these levels depend on characterisic volumes by means of the

next equations:

( )min 1w o pV V V = +

( )max 1 maxw o p uV V V V = + + (51)

From the isothermic transformation law, applied to the air cushion, itresults:

( ) ( )( ) ( )

min min

max max max

H o p a R f

H o p u a R f H

V V V p g H z

V V V V p g H z W

+ + =

= + + + =

(52)

4. CONCLUSIONS

The above shown mathematical model is useful for the next applications:


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1. To assess the main energetical and efficiency parameters for the waterpumping process, that is eSPE and SP .

2. To assess the control levels for pumps startings and stoppings, that is:min / max .

3. To create a computer program for the digital simulation of pumpsfunctioning.

R E F E R E N C E S

1.Alexandrescu O., Staii de pompare. Ed. Gh. Asachi Iai, 2003, ISBN 973-621-059-6.2. Bartha I., Luca M., Popescu t., Popia A. Hidraulica. Culegere de probleme. I.P. "Gh.

Asachi ", Iai, 1992.3. Popescu t. Aplicaii informatice n hidraulica sistemelor hidrotehnice. Editura CERMI, Iai,1999, ISBN 973-8000-11-44. Popescu t., Poiat T. Determinarea caracteristicilor energo-economice echivalente ale unorinstalaii de pompare utilate cu agregate de pompare nereglabile. n: Lucrrile ConferineiNaionale de Termotehnic, Ediia a VII-a, Vol.III, Braov,1997, ISBN 973-97758-5-3.

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