CE Database subject headings: Laboratory tests; Field tests; Simulation; Water distribution systems; Hydraulic models.
Author keywords: Laboratory tests; Field tests; Pressure-discharge relation; Head-driven simulation; Water distribution systems.
Introduction
Methods of hydraulic simulation of water distribution networksare sometimes divided into two groups: demand-driven simulationmethod and head-driven simulation method. In demand-drivensimulation method, it is assumed that available discharge in de-mand nodes is always equal to the required discharge in thosenodes, and under any circumstances the network is able to providethe required demand. This is due to assumption of independencebetween flow and pressure in these nodes. The demand-driven sim-ulation method is applicable only under normal conditions becausein abnormal conditions, such as pipe failure or excess of demand, itis unable to produce realistic outputs. The head-driven simulationmethod is based on assumption of pressure dependency ofoutflows, and therefore has more realistic results than thedemand-driven simulation method under abnormal conditions(Tabesh 1998; Tanyimboh et al. 2001; Tabesh et al. 2002).
A number of pressure-discharge relations (Germanopoulos 1985;Wagner et al. 1988; Fujiwara and Ganesharajah 1993; Gupta andBhave 1996) have been proposed and used in models based onhead-driven simulation method. These equations attempted to re-present the relationship between flow at a model node and pressure
at that node. Blind usage of these relations, which have beenroposed base on some theoretical concepts, is not reasonable andthey need to be validated experimentally. In this study, pressure-discharge curves are obtained, based on the outflow discharge datafrom various faucets for different hydraulic pressures collected fromlaboratory and field measurements to compare flow at a faucet to thepressure at that faucet. Then, performance of the available pressure-discharge relations is investigated using the collected data. The re-search of this paper is a step into validation of the pressure-dischargerelations and can be certainly more completed in the future.
Pressure-Discharge Relations
In the last few years, various relationships have been presentedto express the relation between the available discharge and nodalpressure. These relationships are divided into two categories ofdiscontinuous and continuous relationships. The early equationsexpressing this relationship (Bhave 1981), fall in the discontinuousrelations category [Fig. 1(a)]. In this group, a [0,1] concept is usedto express the pressure-outflow relation in which there is no dis-charge below the minimum required head in each node and fulldemand is available for heads higher than the minimum requiredvalue. Tabesh (1998) and Tabesh et al. (2002) stated that these re-lations cannot be a good demonstrator of the relationship betweenoutflow discharge and pressure at a demand node.
Continuous relations attempt to consider the relationshipetween pressure and outflow discharge for the entire variation do-main continuously (i.e., from zero to the minimum required valueof head). The followings are some of the proposed equations for thecontinuous pressure-discharge relationships.
Germanopolous (1985) proposed the following formula to cal-culate the available outflows at demand nodes [see also Fig. 1(b)]
Qavlj ¼ Qreq
j for Hj ≥ Hdesj ;
Qavlj ¼ Qreq
j
�1 − bj exp
�−cj
�Hj −Hmin
j
Hdesj −Hmin
j
���
for Hminj < Hj < Hdes
j ; Qavlj ¼ 0 for Hj ≤ Hmin
j ð1Þ
1Ph.D. Candidate, School of Civil Engineering, College of Engineering,Univ. of Tehran, P.O. Box 11155-4563, Tehran, Iran (correspondingauthor). E-mail: [email protected]
2Professor, Center of Excellence for Engineering and Management ofCivil Infrastructures, School of Civil Engineering, College of Engineering,Univ. of Tehran. E-mail: [email protected]
3Senior Lecturer, College of Engineering, Mathematics, and PhysicalSciences, Univ. of Exeter, Exeter EX4 4QF, UK. E-mail: [email protected]
4Associate Professor, School of Civil Engineering, Univ. of Urmia,P.O. Box 165, Urmia, Iran. E-mail: [email protected]
j = required dis-charge at node j; Hj = head at node j; Hmin
j = minimum absolutehead at node j, so that if the head at a node is equal to or smallerthan this value, there will be no discharge (where enough informa-tion is not available, the value of Hmin
j is considered equal to theelevation of node); Hdes
j = minimum required head at node j,so that for the heads lower than Hdes
j there is not enoughcapacity to supply the entire required demand [Hdes
j is usuallyconsidered between 147.062 and 294.124 kPa (15 and 30 m) abovethe ground level, depending on the number of floors in eachbuilding]; and bj and cj are empirical coefficients and their valuesare considered as 10 and 5, respectively. According to Eq. (1),if Hj ¼ Hdes
j , then Qavlj ¼ 0.932Qreq
j which is not in linewith expected value of Qavl
j ¼ Qreqj . On the other hand when
Hj −Hminj ¼ 0.46ðHdes
j −Hminj Þ, this equation results in no out-
flow from the node. These issues raise questions about the continu-ity of this relationship, which does not represent actual behavior ofpressure-discharge relation in demand nodes. Tabesh (1998) alsohighlighted that using this relationship in hydraulic analysis leadsto high values for nodal heads.
Gupta and Bhave (1996) modified Eq. (1) as follows[Fig. 1(c)]:
Qavlj ¼ Qreq
j for Hj ≥ Hdesj ;
Qavlj ¼ Qreq
j
"1 − 10
−cj�
Hj−Hminj
Hdesj
−Hminj
�#
for Hminj < Hj < Hdes
j ; Qavlj ¼ 0 for Hj ≤ Hmin
j ð2Þ
This correction addressed some weaknesses of Eq. (1)(Tanyimboh and Tabesh 1997). However, when Hj ¼ Hdes
j , smallchange in the value of cj could substantially decrease the value ofQavl
j in comparison with Qreqj which is not realistic (Tabesh 1998).
Wagner et al. (1988) and Chandapillai (1991) proposed thefollowing formula [Fig. 1(d)]:
Qavlj ¼ Qreq
j for Hj ≥ Hdesj
Qavlj ¼ Qreq
j
�Hj −Hmin
j
Hdesj −Hmin
j
�ð1=mÞ
for Hminj < Hj < Hdes
j Qavlj ¼ 0 for Hj ≤ Hmin
j ð3Þ
where m varies between 1.5 and 2 (Tabesh 1998). In fact, Eq. (3) isa representation of the orifice relationship that is written in a differ-ent form. The orifice relationship can be written as follows:
Qavlj ¼ Qreq
j for Hj ≥ Hdesj Qavl
j ¼ KðHj −Hminj Þn
¼�
Qreqj
ðHdesj −Hmin
j Þn�ðHj −Hmin
j Þn
for Hminj < Hj < Hdes
j Qavlj ¼ 0 for Hj ≤ Hmin
j ð4Þ
where n is usually considered equal to 0.5.Fujiwara and Ganesharajah (1993) introduced the following
relationship between pressure and discharge in demand nodes[Fig. 1(e)]:
Qavlj ¼ Qreq
j for Hj ≥ Hdesj
Qavlj
Qreqj
¼
264RHj
HminjðHj −Hmin
j ÞðHdesj −HjÞdHRHdes
j
HminjðHj −Hmin
j ÞðHdesj −HjÞdH
375
for Hminj < Hj < Hdes
j Qavlj ¼ 0 for Hj ≤ Hmin
j ð5Þ
In this equation, the area under the curve is calculated fromthe minimum absolute head to the available nodal head and is
Fig. 1. Existing pressure-discharge relationships: (a) Bhave (1981); (b) Germanopoulos (1985); (c) Gupta and Bahve (1996); (d) Wagner et al. (1988);(e) Fujiwara and Ganesharajah (1993)
divided by the total area under the curve. This relationship hasnot gained much popularity because of its analytical complexity(Tabesh 1998).
The main difference between Eqs. (3) and (5) is that in Eq. (3)for lower heads, available outflow increases sharply, while, inEq. (5) the sharp increase of outflow relative to its previous statehappens near Hdes
j .More recently Tanyimboh and Templeman (2010) presented the
following logit relationship:
Qavlj ¼ Qreq
j for Hj ≥ Hdesj
Qavlj ¼ Qreq
j
expðαj þ βjHjÞ1þ expðαj þ βjHjÞ
for Hminj < Hj < Hdes
j
Qavlj ¼ 0 for Hj ≤ Hmin
j ð6Þ
in which, parameters αj and βj can be determined via calibration ofrelevant field data. In the absence of field data, these parameters canbe calculated from the following relationships:
αj ¼−4.595Hdes
j − 6.907Hminj
Hdesj −Hmin
jð7Þ
βj ¼11.502
Hdesj −Hmin
jð8Þ
They stated that this formula has the advantages of simplicityand ease of incorporation into the nodal continuity equations and,unlike the other pressure-discharge relations [Eqs. (1)–(3)], thederivative of this equation has no discontinuity at Hj ¼ Hdes
jand Hj ¼ Hmin
j , which is a significant factor in the computationalsolution of the system of equations.
Methodology
For experimental investigation of the pressure-discharge relation, alaboratory set up was built as shown in Fig. 2. A single-speed pumpwas used to produce a constant head. Faucets (1) and (2) were alsoused to control the pressure behind Faucet (3), where flow dis-charge was measured. A loop system was created by introducingtwo tanks to prevent excessive use of water during the experiment.The elevations of manometer and Faucet (3) were the same and thedistance between them was about 15 cm. The pipe diameters usedin this set up were 1=2 and 3=4 in. During this experiment, Faucet(3) was set open in four positions (completely open, half open, andone fourth and one eighth of full opening) and the measurements ofpressure and outflow were made.
In addition, in a field measurement program, available outflowdischarge data in three different points (Points 1, 2, and 3) of thewater distribution network of Urmia city with different hydraulicpressure values were collected. Urmia is a city located in thenorth west of Iran, with 1,289 km water distribution network and147,475 properties. Points 1 and 2 were houses and Point 3 was astore. In these points the elevations of manometer and faucet werethe same and the distance between them was about 15 cm withoutany water use. The diameter of the pipe on which the faucet wasinstalled was 1=2 in. To investigate the effects of faucet type onoutflow discharge, three types of faucets were used: a half-inchsimple faucet, a three-quarter-inch simple faucet, and a water-saving faucet.
To evaluate accuracy of the pressure-discharge relations, thefollowing indicators of root mean squared error (RMSE), normal-ized mean square error (NMSE), mean absolute error (MAE), andthe coefficient of determination (R2) are used
where Qobs = observed outflow discharge; varðQobsÞ = variance ofthe observed outflow discharges; Q̄obs = mean of the observed out-flow discharges; Qcal = calculated outflow discharge; Q̄cal = meanof the calculated outflow discharges; and I = number of data points.
Results and Discussion
Figs. 3 and 4 show the curves of observed outflow discharge froma half-inch faucet in fully open status in different locations of theUrmia water distribution network and also the laboratory set upfor various static and dynamic hydraulic pressures (Ps and Pd).In this paper, static and dynamic hydraulic pressures were consid-ered as the pressures behind the faucet in closed and open states,respectively. These figures show that the outflow dischargeincreases as the pressure increases and this trend is continued
up to pressure of 784.331 kPa (80 m). This is contrary to pressure-discharge relations presented so far in which a fixed discharge isconsidered for pressures higher than the minimum required value.Another point is that the gradient of the outflow discharge-pressurecurve is higher near the minimum absolute pressure than the gra-dient in higher pressures. From Figs. 3 and 4, it can be seen thatalthough the amount of outflow discharge per certain hydraulicpressure varies in different locations of the network, but the shapeof outflow discharge-pressure curves in different locations of thenetwork is the same.
Variations of the observed outflow discharge from various fau-cets in completely open status in different static hydraulic pressuresat Point 3 are illustrated in Fig. 5. Although this figure shows thatoutflow discharge for a certain hydraulic pressure varies dependingon the type of faucet, the curves of different faucets have the sameshape and almost follow a specific relationship. Fig. 5 confirmsthe performance of water-saving faucets and their role in demand-management schemes. As can be seen in this figure, the outflow ofthe water-saving faucet is about half of the outflow discharge of thesimple conventional faucet.
To calculate the amount of outflow discharge using differentpressure-outflow relations, Hdes
j is considered equal to 20 or30 m and Hmin
j equal to zero. Then the desired pressure(Pdes
j ¼ Hdesj −Hmin
j ) will be equal to 20 or 30 m, respectively.The desired pressure is the minimum required pressure, so thatfor the values lower than it there is not enough capacity to supply
the entire required demand. The performances of the pressure-discharge relations have been investigated in two ways:1. The third interval of the pressure-discharge relationship
(i.e., the range Hj ≥ Hdesj ) has not been used and the outflow
discharge for heads greater than Hdesj is calculated using
the second interval of the pressure-discharge relationship(Hmin
j < Hj < Hdesj ). For example in this mode the pressure-
discharge relation proposed by Wagner et al. (1988) will beas below:
Qavlj ¼ 0 for Hj ≤ Hmin
j
Qavlj ¼ Qreq
j
�Hj −Hmin
j
Hdesj −Hmin
j
�ð1=mÞfor Hj > Hmin
j ð13Þ
2. Complete form of the pressure-discharge relationship hasbeen used.
The parameters αj and βj in the relationship proposed byTanyimboh and Templeman (2010) are calibrated using availablefield data and their values are determined as (−1.513) and (0.232)respectively. To calculate the outflow discharge using this relation-ship, the calculated values of the parameters αj and βj by usingEqs. (7) and (8) and also their calibrated values have been usedas Cases 1 and 2, respectively, from Tanyimboh and Templeman(2010). The obtained results in these cases are shown in Tables 1and 2 and also in Figs. 6–8.
The performances of pressure-discharge relations for differentvalues of desired pressure (Pdes
j ¼ 20 or 30 m) have been evaluatedat Point 3 using the RMSE, NMSE, MAE, and R2 criteria and theaverage values of these criteria are shown in Table 1. The resultsshow that the orifice and Wagner relations better matched the ex-perimental data. The orifice and Wagner relations in Mode 1, wherethe third interval of these relations is omitted and the outflow fol-lows the middle interval term, show better performance. Also it canbe seen that the performance of Tanyimboh and Templeman rela-tion Eq. (6) is improved in the case of using the calibrated valuesof parameters αj and βj. It should be noted that since the results ofother points were similar to that of Point 3, therefore only some ofthe obtained results for Points 1 and 2 are shown in Table 2. As canbe seen in this table, the results of other points consistently showthat the orifice and Wagner relations have better performance.
Figs. 6–8 show comparison of the calculated pressure-dischargecurves obtained by using the pressure-discharge relationships withthose of observed curves for fully open half-inch, three-quarter-inch and water-saving faucets in Mode 1 at Point 3, respectively.It is observed that the calculated pressure-discharge curves
Fig. 3. Observed outflow discharge values and static hydraulicpressures for half-inch faucet in fully open status from laboratory(solid diamonds) and field measurements at Point 1 (solid squares),Point 2 (open triangles), and Point 3 (crosses)
Fig. 4. Observed outflow discharge values and dynamic hydraulicpressures for half-inch faucet in fully open status from laboratory (soliddiamonds) and field measurements at Point 1 (solid squares), Point 2(open triangles), and Point 3 (crosses)
Fig. 5. Observed outflow discharge values and static hydraulicpressures for different faucets in fully open status at Point 3: half-inch(solid diamonds), three-quarter inch (solid squares), and water saving(open triangles)
obtained by the orifice and Wagner relations are very similar to theobserved curves.
A New Pressure-Discharge Relationship
The orifice formula was considered for further investigation as it per-formed better than all of the existing pressure-discharge relations. It
was used to calculate available outflow discharges in laboratory andalso in different points of the water distribution network for differentvalues of desired pressure and pressure exponent.
Fig. 9 shows the pressure-discharge curves calculated using theorifice formula and observed pressure-discharge curves for fullyopen half-inch faucet at Point 3. This figure shows that the calcu-lated pressure-discharge curves are different for different values ofthe desired pressure but have a similar shape.
Table 1. Average Values of Evaluation Criteria for Various Pressure-Discharge Relationships in Different Modes for the Half-Inch, Three-Quarter-Inch andWater-Saving Faucets at Point 3
Table 2. Average Values of Evaluation Criteria for Various Pressure-Discharge Relationships in Different Modes for the Half-Inch Faucet at Points 1 and 2
Fig. 6. Comparison of the observed (solid circles) and calculatedpressure-discharge curves for fully open half-inch faucet and Pdes ¼30.0 m in Mode 1 at Point 3: Eq. (2) (open squares), Gupta and Bhave(1996); Eq. (4) (solid diamonds), orifice; Eq. (5) (open triangles),Fujiwara and Ganesharajah (1993); Eq. (13) (solid squares), Wagneret al. (1988); Eq. (6) Case 1 (solid triangles) and Case 2 (asterisks),Tanyimboh and Templeman (2010)
Fig. 7. Comparison of the observed (solid circles) and calculatedpressure-discharge curves for fully open three-quarter-inch faucetand Pdes ¼ 20.0 m in Mode 1 at Point 3: Eq. (2) (open squares), Guptaand Bhave (1996); Eq. (4) (solid diamonds), orifice; Eq. (5) (open tri-angles), Fujiwara and Ganesharajah (1993); Eq. (13) (solid squares),Wagner et al. (1988); Eq. (6) Case 1 (solid triangles) and Case 2(asterisks), Tanyimboh and Templeman (2010)
The minimum value of MAE is related to the pressureexponent of 0.48. Therefore, the most appropriate amount forpressure exponent in the orifice relation is considered as 0.48 inthis paper.
The available outflow discharge from different faucets in differ-ent states and locations for different values of desired pressure arecalculated using the orifice formula with pressure exponent valueequal to 0.48. Table 3 presents the desired pressure values that leadto the minimum values of the evaluation criterion. The appropriatevalue for desired pressure at each point is close to the maxi-mum pressure in the network at that point. For example, for thelaboratory set up in which the maximum measured pressure was67.5 m, the appropriate value of the desired pressure was obtainedas 40 or 50 m. At Point 3 that the maximum measured pressure wasequal to 80 m, the appropriate values of the desired pressure wereobtained as 60 or 70 m for different types of faucets. Accordingto Table 3, the average value of desired pressure will be equalto 53.75 m which can be considered as appropriate value for thedesired pressure to be used for faucets in different parts of thenetwork.
It is important to note that the usage of the term desired pressurein this paper, follows a similar term that is used in the most ofpressure-discharge relations presented so far and is defined as apressure above which the required demand is fully supplied. How-ever, in most standard codes such as IRIVPSPS (2011), a value of50 m is considered as the maximum allowed pressure to prevent thepipe burst and high leakage.
If the maximum allowed pressure is used, the modified form ofthe orifice relationship will be as follows:
Qavlf ¼
8>>>>>><>>>>>>:
0 if Pf ≤ 0
kfP0.48f ¼ Qavlð50Þ
f
ðPalwmaxÞ0.48
P0.48f if 0 < Pf ≤ Pthres
kfðPthresÞ0.48 ¼ Qavlð50Þf
ðPalwmaxÞ0.48
ðPthresÞ0.48 if Pthres < Pf
ð14Þ
where Qavlf = available outflow discharge at faucet; Qavlð50Þ
f =available outflow discharge at faucet for pressure of 50 m;kf = coefficient of pressure at faucet; Pf = pressure at faucet;Palwmax = maximum allowed pressure (50 m) used to calculate
the pressure coefficient; and Pthres = threshold pressure abovewhich the outflow discharge will be constant. When pressure risesto the desired value, 100% of the required demand will be supplied.The outflow discharge may keep increasing as the pressureincreases. However, the outflow discharge is not affected by pres-sure if the pressure is above a threshold. There is such thresholdeffect for most demands except for leakage, which continuouslyincreases with pressure (Wu et al. 2009). The Pthres can be consid-ered about 100 m, because according to the observed pressure-discharge curves, for pressures higher than 80 m, the gradientof these curves is decreased. On the other hand, the maximumvalue of hydraulic pressure in the case study was about 80 mand it was not possible to measure the outflow discharge for pres-sures higher than 80 m to obtain the exact value of the thresholdpressure. Therefore, it is assumed that for pressures higher than100 m the gradient of the curve is very low and the outflow dis-charge is constant. By substituting the maximum allowed pressureand the threshold pressure in Eq. (14) the following equation isobtained:
Qavlf ¼
8>>>>>><>>>>>>:
0 if Pf ≤ 0
kfP0.48f ¼ Qavlð50Þ
f
ð50Þ0.48 P0.48f ¼ Qavlð50Þ
f
6.539P0.48f if 0 < Pf ≤ 100 m
kfð100Þ0.48 ¼Qavlð50Þ
f
ð50Þ0.48 ð100Þ0.48 ¼ 1.395Qavlð50Þ
f if Pf > 100m
ð15Þ
Fig. 8. Comparison of the observed (solid circles) and calculatedpressure-discharge curves for fully open water-saving faucet andPdes ¼ 30.0 m in Mode 1 at Point 3: Eq. (2) (open squares), Guptaand Bhave (1996); Eq. (4) (solid diamonds), orifice; Eq. (5) (opentriangles), Fujiwara and Ganesharajah (1993); Eq. (13) (solid squares),Wagner et al. (1988); Eq. (6) Case 1 (solid triangles) and Case 2(asterisks), Tanyimboh and Templeman (2010)
Fig. 9. Variations of the observed (solid circles) and calculatedoutflows obtained by the orifice relation considering different desiredpressures, plotted by static pressure for fully open half-inch faucet atPoint 3
Thus, according to this relationship the available outflow willcontinue to increase for pressures higher than Palw
max ¼ 50 m, butafter reaching to the value of 100 m (i.e., Pthres) the amount ofoutflow remains constant.
To calculate the available outflow discharge using this relation-ship, the values of existing pressure and available outflow for themaximum allowed pressure (50 m) are needed. The value of avail-able outflow corresponding to pressure of 50 m varies in differentlocations of the network and depends on the characteristics andconditions of network. However, according to the standard codessuch as IRIVPSPS (2011), in the design and hydraulic analysis ofwater distribution networks, water demands are known and pres-sures of 15–30 m is considered as the desired pressure (Pdes) orminimum allowed pressure (Palw
min). This means that for pressuresof 15–30 m (depending on number of floors), the water demandis supplied completely, and in fact the available outflow dischargeis equal to the required demand. Considering the minimum allowedpressure (30 m), the appropriate value for pressure exponent canbe considered equal to 0.51. Therefore, the available outflow dis-charge in this pressure (required demand) can be used in theterm Qavlð30Þ
f =ðPalwminÞ0.51. Thus the final revised form of the orifice
relation will be as follows:
Qavlf ¼
8>>>>>><>>>>>>:
0 if Pf ≤ 0
kfP0.51f ¼ Qavlð30Þ
f
ðPalwminÞ0.51
P0.51f if 0 < Pf ≤ Pthres
kfðPthresÞ0.51 ¼ Qavlð30Þf
ðPalwminÞ0.51
ðPthresÞ0.51 if Pf > Pthres
ð16Þ
or, using a Palwmin of 30 m and Qavlð30Þ
f ¼ Qreqf .
Qavlf ¼
8>>>>>><>>>>>>:
0 if Pf ≤ 0
kfP0.51f ¼ Qavlð30Þ
f
ð30Þ0.51 P0.51f ¼ 0.176ðQreq
f × P0.51f Þ if 0 < Pf ≤ 100 m
kfð100Þ0.51 ¼Qavlð30Þ
f
ð30Þ0.51 ð100Þ0.51 ¼ 1.848Qreq
f if Pf > 100 m
ð17Þ
The amount of available outflow discharge from faucets indifferent locations of the network is calculated using the modifiedorifice relations in Eqs. (15) and (17). Some of the calculatedpressure-discharge and observed curves are given in Figs. 10–13.In these figures the Calculated Discharge (I) refers to the availabledischarge values calculated by Eq. (15) and Calculated Discharge(II) refers to the available discharge values calculated by Eq. (17).The figures show that the modified orifice relations [Eqs. (15)and (17)] have an acceptable performance in estimating availableoutflow discharge, while according to Figs. 6–8, the existingpressure-discharge relations were not able to calculate the outflowvalues accurately and the calculated flows by those relations weresignificantly different from the observed ones.
As Giustolisi and Walski (2011) demonstrated, demands inwater distribution systems are divided into two types, volumetricand pressure-dependent demands. The pressure-dependent de-mands are also classified into three types including human-baseddemand, uncontrolled orifice-based demand, and leakage-baseddemand. In other words, urban water consumption according tothe various standard codes are divided into various categories such
as household, industrial, public, and green space; and, for everycategory a range of appropriate values are proposed. A portion ofthis consumption, like consumption of the dishwasher, washingmachines, tank style toilets, industrial process tanks, and bath tubshave specified volume, which after supplying the required volume,no more outflow is required. This consumption which has a knownvolume is called volumetric consumption and the related outflowsare called volumetric outflow. Another part of the urban consump-tion including human-based demand (like faucets of all the wash-basins and showers), uncontrolled orifice-based demand (like fireprotection systems, sprinkler systems, and landscape irrigation sys-tems), and leakage-based demand (like background leakage andreported and unreported bursts) are changed related to the pressure.This consumption is called pressure-dependent consumption andthe related outflow discharges are called pressure-dependent out-flows [see Giustolisi and Walski (2011) for further details]. Someresearches, such as Giustolisi and Walski (2011), have tried to de-termine the exact proportion of these two types of consumptions;however, as pointed out in their paper, more research is still neededin this regard.
Table 3. Desired Pressure Values for Using in Orifice Formula withPressure Exponent Equal to 0.48
Appropriatevalues ofdesiredpressure(m) Faucet mode Faucet type Faucet position
40 Fully open Half-inch faucet Laboratory(maximum
pressure = 67.5 m)40 Half open50 One quarter open50 One eighth open50 Fully open Half-inch faucet Point 1 (maximum
pressure = 55 m)50 Half open50 Fully open Half-inch faucet Point 2 (maximum
pressure = 53.5 m)50 Half open70 Fully open Half-inch faucet Point 3 (maximum
pressure = 80 m)10 Half open60 One quarter open70 One eighth open70 Fully open Three-quarter-inch
In order to further improve the accuracy of Eq. (17), anothermodification is needed to take into account the impact of volumet-ric outflow. In Eq. (17), the outflow discharge is considered aspressure-dependent outflow. Then by applying the volumetric partof outflows in Eq. (17), the following formula results:
Qavlf ¼
8>>>>><>>>>>:
0 if Pf ≤ 0
0.176ðQreqf × P0.51
f Þ if 0 < Pf ≤ 30 m
a×Qreqf þ 0.176ðb×Qreq
f × P0.51f Þ if 30 < Pf ≤ 100 m
a×Qreqf þ 1.848ðb×Qreq
f Þ if Pf > 100 m
ð18Þ
which can be rearranged
Qavlf ¼
8>>>>><>>>>>:
0 if Pf ≤ 0
0.176ðQreqf × P0.51
f Þ if 0 < Pf ≤ 30 m
Qreqf ðaþ 0.176b × P0.51
f Þ if 30 < Pf ≤ 100 m
Qreqf ðaþ 1.848bÞ if Pf > 100 m
ð19Þ
where a and b are the proportion of the volumetric and pressure-dependent consumption, respectively (aþ b ¼ 1). Eq. (19) is sug-gested as an appropriate pressure-discharge relation (for mostdemands, except leakage-based demand and uncontrolled orifice-based demand like sprinkler systems) to be used in models based onhead-driven simulation method. As can be seen in Fig. 14, thecalculated available outflow will be changed by the variation ofparameters a and b. For example, if the volumetric outflow isconsidered as 50% of the total discharge and the remaining 50%is regarded as pressure dependent (a ¼ b ¼ 0.5), then Eq. (19) willbe written
Qavlf ¼
8>>>>><>>>>>:
0 if Pf ≤ 0
0.176ðQreqf × P0.51
f Þ if 0 < Pf ≤ 30 m
Qreqf ð0.5þ 0.0882P0.51
f Þ if 30 < Pf ≤ 100 m
1.424Qreqf if Pf > 100 m
ð20Þ
According to Eq. (20), for pressures greater than 30 m, 50% ofoutflow is volumetric and will remain constant and the other part
Fig. 10. Variations of the observed (solid circles) and calculated out-flow discharge using modified orifice formulae for half-open half-inchfaucet at Point 1; Calculated Discharge (I) (open diamonds) was deter-mined using Eq. (15) and Calculated Discharge (II) (solid squares) wasdetermined using Eq. (17)
Fig. 11. Variations of the observed and calculated outflow dischargeusing modified orifice formulae for half-open half-inch faucet atPoint 2; Calculated Discharge (I) (open diamonds) was determinedusing Eq. (15) and Calculated Discharge (II) (solid squares) was de-termined using Eq. (17)
Fig. 12. Variations of the observed and calculated outflow dischargeusing modified orifice formulae for fully open half-inch faucet atPoint 3; Calculated Discharge (I) (open diamonds) was determinedusing Eq. (15) and Calculated Discharge (II) (solid squares) was de-termined using Eq. (17)
Fig. 13. Variations of the observed and calculated outflow dischargeusing modified orifice formulae for fully open water-saving faucet atPoint 3; Calculated Discharge (I) (open diamonds) was determinedusing Eq. (15) and Calculated Discharge (II) (solid squares) was de-termined using Eq. (17)
of the outflow is pressure-dependent and increases as the pressureincreases until the pressure reaches 100 m. For pressures higherthan 100 m, the total amount of outflow discharge will be consid-ered as volumetric and will remain constant. It should be noted thatthere really is no threshold pressure, but one is used as a matter ofconvenience to keep reasonable water use in models. In reality,what would happen at very high flows is that as pressure at themodel node increases, flow would continue to increase, but even-tually head loss in service lines and water meters would be so greatthat pressure at the faucet would not increase as node pressure in-creased and hence flow would eventually not increase. In addition,for a given use such as hand washing, the water user would notnecessarily open the orifice fully but would limit the flow to somedesirable value. Another point is that there really is no sharptransition between parts of the pressure-discharge equation butthe authors feel that 30 and 100 m are reasonable values forthis paper.
Conclusion
In this study, the performance of existing pressure-discharge rela-tions was investigated by experimental and field measurements ofavailable outflows from different faucets under various hydraulicpressures. A new pressure-discharge relation was presented thatis more reasonable than some previous relationships. From the re-sults of this study, the following conclusions can be drawn:1. Generally, for an individual faucet with a fixed opening, the
rate of outflow discharge increases with increasing pressure,and this trend continues for pressures higher than the mini-mum desired value (e.g., 30 m); however, for human- andvolume-related types of demand in many real cases, as pres-sure increases, users do not open the faucets as wide;
2. While the experiments relate flow at a faucet to pressure atthat faucet, the relationship between flow at a node and pres-sure at that node is more complicated because of the variablehuman behavior of the person opening the faucet, the differ-ence in elevations of the faucet and the model node, and thecomplexity of piping between them;
3. Although the outflow discharge at a certain hydraulic pressuredepends on the type of faucet, the pressure-discharge curves ofdifferent faucets have the same basic shape, as predicted by theorifice equation; and
4. The pressure-discharge relation presented by Wagner et al.(1988) better matched experimental data, because it closelyresembled the orifice relation.
Acknowledgments
The authors would like to acknowledge the financial supportof University of Tehran for this research under grant number8102050/1/04. The comments and suggestions by two reviewersare also gratefully acknowledged for improving the paper.
Notation
The following symbols are used in this paper:a = proportion of volumetric outflow;b = proportion of pressure-dependent outflow;
bj, cj = empirical coefficients used in the pressure-dischargerelation;
Hj = head at node j;Hdes
j = minimum required head at node j;Hmin
j = minimum absolute head at node j;I = total number of data points;
kf = coefficient of pressure at the faucet;kj = coefficient of pressure at node j;
1=m = head exponent in pressure-discharge relationproposed by Wagner et al. (1988);
MAE = mean absolute error;n = pressure exponent in Orifice formula;
NMSE = normalized mean squared error;Palwmax = maximum allowed pressure;
Palwmin = minimum allowed pressure;
Pdes = desired pressure;Pf = pressure at the faucet;Pj = pressure at node j;
Qcal = calculated outflow discharge;Qavl
f = available discharge at the faucet;
Qavlð30Þf = available outflow discharge at the faucet for pressure
of 30 m;Qavlð50Þ
f = available outflow discharge at the faucet for pressureof 50 m;
Qavlj = available discharge at node j;
Qobs = observed outflow discharge;Qreq
j = required discharge at node j;
Q̄cal = mean of the calculated outflow discharges;Q̄obs = mean of the observed outflow discharges;R2 = coefficient of determination;
RMSE = root-mean squared error;varðQobsÞ = variance of observed outflow discharges; and
αj, βj = parameters used in pressure-discharge relation.
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