Pie Netwrks - Hardy Crss etd Page Pie Netwrks Itrducti A ie etwrk is a itercected set f ies likig e r mre surces t e r mre demad (delivery) its, ad ca ivlve ay umber f ies i series, bracig ies, ad arallel ies. Witi te ctext f civil egieerig alicatis, ie etwrks iclude muicial water distributi systems ad um stati frce mais. Tyical calculatis iclude sizig ies ad/r ums, determiig te directi ad magitude f flw i ies, ad determiig ressures (HG) at des. Pie etwrks are geerally t cmlicated t slve aalytically. Accrdigly, we resrt t cmuter rgrams suc as WaterCAD, a rgram develed ad marketed by Haestad etds, Ic. Oter rgrams yu may ecuter i rfessial ractice are CyberNET (earlier AutCAD versi f WaterCAD), EPANET (develed ad available frm te US Evirmetal Prtecti Agecy), ad YPIPE (PIPE000) (develed at te Uiversity f etucky ad available trug its Civil Egieerig Sftware Ceter). Csider te system illustrated i te fllwig scematic diagram. ECIV 36 Itrducti t Water Resurces Egieerig Deartmet f Civil ad Evirmetal Egieerig, Uiversity f Sut Carlia.
Pie Netwrks - Hardy Crss etd Page Te strage tak (T-) ad tree demad des (J-, J-4 ad J-5) are sulied by a well (mdeled as a reservir, R-, ad um, PP-) trug a system f te ies (P-, P-,, P-0) tat frm a etwrk wit tw rimary ls ad e at. A rimary l is a clsed l tat ctais ter ls. Fr tis system, ies P-, P-6, P-7 ad P-8 frm a rimary l. A at is a series f ies cectig tw fixed grade des (FGNs), wic are its f kw ttal eergy. Fr tis system, te well (reservir R-) ad tak are fixed grade des. Altug tere are may cmbiatis f ies i series tat cect R- ad T-, regardless f te at fllwed, te ead lss is te same. Csequetly, tere is ly e ydraulic at, wic may be rereseted by ay set f ies i series cectig R- ad T-. As a examle, ies P-9, P-0, P-7 ad P- rereset e way t exress te at betwee R- ad T-. Flw i ay ie etwrk must satisfy te fllwig rules (cditis):. Te flw it ay jucti must equal te flw ut f it (dal ctiuity).. Te eergy balace alg a ie r series f ies must satisfy te cservati f eergy ricile as stated by te Berulli equati. Eergy lss i ay ie r series f ies is due t ie fricti ad mir lsses. 3. Te algebraic sum f ead lsses arud ay clsed (rimary) l must be zer. Hardy Crss etd Oe ractical arac fr ad (sreadseet) sluti f flws i small ie etwrks is te metd f successive arximatis, itrduced i 936 by Hardy Crss. Altug rigially develed fr led systems, it is readily exaded t iclude systems wit ats. Tis metd uses a iterative rcedure t adjust te flw i all ies arud eac rimary l util te ead lsses sum t zer, ad i all ies alg eac at util te eergy equati is satisfied. Te eart f tis metd lies i te determiati f te l (r at) flw crrecti,. Primary T devel te equatis t determie fr rimary ls, start by writig ead lss alg eac ie i te fllwig geeral frm () i wic () were is te crrect discarge, is te assumed discarge, ad is te crrecti. Substitute fr i te ead lss exressi ad erfrm te exasi t btai ECIV 36 Itrducti t Water Resurces Egieerig Deartmet f Civil ad Evirmetal Egieerig, Uiversity f Sut Carlia.
Pie Netwrks - Hardy Crss etd Page 3 ECIV 36 Itrducti t Water Resurces Egieerig Deartmet f Civil ad Evirmetal Egieerig, Uiversity f Sut Carlia. ( ) ( ) ( ) ( ) (3) If is small cmared t, te terms f te series after te secd term may be eglected. ( ) ( ) (4) Fr a rimary l wit te same i all ies, ( ) ( ) ( ) 0 (5) Slve tis equati t btai te fllwig exressi fr fr rimary ls. ( ) (6) Furter examiati f (6) sws te terms i te demiatr are te first derivative f te ead lss terms wit resect t discarge,. ( ) ( ) f f (7) Accrdigly, te exressi fr fr rimary ls als ca be writte as (8) Pat Fr ats, begi by writig te eergy equati betwee tw fixed-grade des. f EG E EG (8) Pum eergy, E, is iut i terms f useful rsewer r by rvidig ifrmati abut te um caracteristic curve. Te um curve is best rereseted wit a equati tat gives um ead as a fucti f discarge, e.g.,
Pie Netwrks - Hardy Crss etd Page 4 E a b C (9) Tere are ter equatis tat ca be used, but tis is e f te mst ular frms. Rearrage te eergy equati t gru all terms tat are fuctis f discarge te leftad side ad te remaiig terms te rigt-ad side t btai ( ) E EG EG EG f f (0) Fllwig similar lgic alied t devel (8), substitute () it (0) ad exress te exasi i derivative terms similar t (7). f E () ( ) E EG Slve () fr te flw crrecti alg a at. EG E E () were EGEG - EG. Geeral Prcedure Te Hardy Crss metd csists f te fllwig elemets.. By careful isecti, assume te mst reasable distributi f flws tat satisfies ctiuity at eac de.. Write ead lss alg eac ie i te frm Fr rimary ls 3. Cmute te algebraic sum f te ead lsses arud eac rimary l. Csider lsses frm clckwise flws as sitive ad lsses frm cuter-clckwise flws are egative. 4. Adjust te flw i eac rimary l by a crrecti,, util te ead lss sums t zer. Fr ats 5. Cmute te algebraic sum f ead lsses alg te series f ies betwee te ustream ad dwstream FGNs. Csider lsses frm flws i te directi f te dwstream FGN as sitive ad lsses frm flws i te directi f te ustream FGN as egative. ECIV 36 Itrducti t Water Resurces Egieerig Deartmet f Civil ad Evirmetal Egieerig, Uiversity f Sut Carlia.
Pie Netwrks - Hardy Crss etd Page 5 6. Adjust te flw i eac ie by a crrecti,, util te ead lss sums t te differece i HG betwee te ustream ad dwstream FGNs. Examle Prblem Give te water distributi etwrk sw i te fllwig scematic. Tis system as 9 ies, 6 des, ad suly taks. Te demads at juctis J-3, J-5 ad J-6 are 50, 00 ad 50 gm, resectively. All ies are CIP (C HW 30) ad,500 feet lg. Te diameter f ies P- ad P-9 is -ices, ad f ies P-, P-3 ad P-4, 8-ices. Te diameter f te ter ies is 6-ices. Te water surface elevati (HG) is 35 feet i tak T- ad 40 feet i tak T-. (a) Determie te magitude (gm) ad directi f flw i eac ie. (b) Determie te HG (feet) ad ressure (si) at eac de. Sluti: Aly te Hardy-Crss metd. Use te Haze-Williams equatis fr ead lss. Ste : First, determie fr eac ie ad use te fllwig geeral exressi fr frictial ead lss. 4.73 i wic.85 fr te Haze-Williams equati ad..85 4. 87 C D f HW ECIV 36 Itrducti t Water Resurces Egieerig Deartmet f Civil ad Evirmetal Egieerig, Uiversity f Sut Carlia.
Pie Netwrks - Hardy Crss etd Page 6 Pie N egt, ft Diameter, C HW i P- 500 30 0.86 P- 500 8 30 6. P-3 500 8 30 6. P-4 500 8 30 6. P-5 500 6 30 5.3 P-6 500 6 30 5.3 P-7 500 6 30 5.3 P-8 500 6 30 5.3 P-9 500 30 0.86 Ste : Assume flw magitude ad directi fr eac ie. Remember t satisfy dal ctiuity. If te iitial distributi f flws is crrect, te ead lsses arud eac rimary l will sum t zer; terwise, tey will t sum t zer ad yu must cmute flw crrectis. Pie N U/S Nde D/S Nde, gm, cfs P- T- J- 00 0.3 P- J- J- 50 0. P-3 J- J-3 00 0.3 P-4 J-3 J-4 50 0.557 P-5 J-4 J-5 50 0.334 P-6 J-5 J-6 00 0.3 P-7 J- J-6 50 0. P-8 J- J-5 50 0.334 P-9 T- J-4 400 0.89 Ste 3: Use wrktables fr tw rimary ls ad e at. Begi wit te assumed flw distributi ad iterate util te sum f ead lsses arud eac rimary l is zer (witi tlerace) ad te ead lsses alg te at equals te differece i HG betwee te tw taks. I Pie - P- 6. 0. 0.06 0.956-0.00 P-8 5.3 0.334 3.3 9.9 0. P-6 5.3 0.3.567 7.05 0.0 P-7 5.3-0. -0.430 3.877-0.4.85 Σ4.553 Σ.770 ( 4.553) (.770) 0.3 ECIV 36 Itrducti t Water Resurces Egieerig Deartmet f Civil ad Evirmetal Egieerig, Uiversity f Sut Carlia.
Pie Netwrks - Hardy Crss etd Page 7 II Pie - P-3 6. -0.3-0.386.73-0.06 P-4 6. -0.557 -.04 3.778-0.540 P-5 5.3 0.334 3.3 9.9 0.35 P-8 5.3-0. -.54 6.97-0.04 Σ-0.7 Σ.393.85 ( 0.7) (.393) 0.07 Pat T- t T- Pie - P- 0.86 0.3 0.053 0.39 0.050 P- 6. -0.00 0.000 0.03-0.75 P-3 6. -0.06-0.333.69-0.379 P-4 6. -0.540 -.987 3.680-0.73 P-9 0.86-0.89-0.694 0.779 -.064 Σ-.96 Σ6.348 ( EG EG ) (.96) ( 5).85( 6.348) 0.73 I Pie - P- 6. P-8 5.3 P-6 5.3 P-7 5.3 Σ Σ II Pie - P-3 6. P-4 6. P-5 5.3 P-8 5.3 Σ Σ ECIV 36 Itrducti t Water Resurces Egieerig Deartmet f Civil ad Evirmetal Egieerig, Uiversity f Sut Carlia.
Pie Netwrks - Hardy Crss etd Page 8 Pat T- t T- Pie - P- 0.86 P- 6. P-3 6. P-4 6. P-9 0.86 Σ Σ Ctiue iteratis util te flw crrecti,, fr all ls ad ats are less ta a rescribed tlerace. Te flw magitudes ad directis, ad dal HG values, tat yu suld btai fr tis examle are sw i te fllwig scematic. Tese results were btaied wit WaterCAD. ECIV 36 Itrducti t Water Resurces Egieerig Deartmet f Civil ad Evirmetal Egieerig, Uiversity f Sut Carlia.