Simple Iterative Model for Adjusting Hazen-Williams Friction Coefficient for Drip Irrigation laterals.

Transcription

1 Australian Journal o Basic and Applied Sciences, 5(12): , 2011 ISSN Simple Iterative Model or Adjusting Hazen-Williams Friction Coeicient or Drip Irrigation laterals. 1 A.A. Alazba and 2 M.B. ElNesr 1 Abdulraman Ali Alazba. Proessor, Al Amoudi Cair or water researces, King Saud University. 2 Moammad, N. BagatElNesr. Assistant Proessor, Al Amoudi Cair or water researces, King Saud University. Abstract: Hazen-Williams equation is used widely by irrigation systems designers due to its simplicity.however, Darcy-Weisbac equation is more accurate and reliable. Te accuracy o te latter is due to its riction coeicient, wic depends on bot low caracteristics and pipe surace state. On te oter and, Hazen-Williams coeicient (C) depends only on pipe substance and age. A comparative analysis o bot models was perormed troug a simple iterative model. Te analysis was based on te real state design procedure o drip laterals. More accurate values o coeicientcwere suggested to be used in designing drip laterals. A straigtorward equation was developed to compute C depending on emitter lowrate, emitter low exponent, and pipe diameter.te results reveal tat C ranges rom 132 to 138 or drip laterals, wile it was proved tat using C=150 is reasonable or maniolds design. Key words: Hazen-Williams, Darcy-Weisbac, Friction Coeicient, Curcill equation, Drip Irrigation laterals design. INTODUCTION Designing a drip irrigation network is an operation in wic pipelines diameters and lengts are determined and optimized or economic and eicient system operation.in practice, te dripper line diameter is predetermined by te manuacturing availability o lines wit built in emitters, tus, most networks ave 16mm outer diameter (OD) polyetylene (PE) dripperlines.in some cases, 13mmOD or 19mm OD dripper lines are used. Te goal or designing dripper laterals is to determine its maximum lengt to ensure acceptable uniormity over te subunit. On te oter and, designing maniolds deals more wit speciyingteir diameters as teir lengts are usually limited by network planning. However, sizing eiter type is mainly based on te riction losses limitation to te allowable amount.friction losses are calculated by several metods. Te most amous metods in irrigation design are Darcy-Weisbac (D-W), and Hazen-Williams (H-W). (Allen, 1996) related D-W and H-W riction actors wit eynolds number, concluding te importance o adjusting H-W riction actor (C) wit canging pipe velocity and diameter. (Mogazi, 1998) conductedsome laboratory experiments to determine te proper values o H-W actor.he reported te values or te commonly used pipe sizes in trickle irrigation, and compared te percent o increase in riction due to ixing te value o C. Sayyaa and Sablani (1998) accomplised an artiicial neural network to calculate te D-W riction actor, teir model agrees very well wit te Colebrook equation in te turbulent stage o te low. (Valiantzas, 2005) compared te H-W and D-W riction actors, and developed a power unction or tis relation. (Martorano, 2006) acieved a comparative study between D-W and H-W, recommending te usage o D-W due to its precision. (Yildrim and Ozger, 2008) developed a Neuro-uzzy approac in estimating H-W riction coeicient or small diameter polyetylene pipes. Tey proved tat ixingh-w coeicient over all PE diameters migt lead to considerable error in riction loss computation. Friction losses calculation metods: Friction losses calculation is most accuratelyperormed by te Darcy-Weisbac equation (Eqn. 1). 2 L v D 2g (1) were :pressure ead loss due to riction (L), : riction actor, l: pipe lengt (L), D: pipe diameter (L), v: water velocity (LT -1 ), and g: acceleration o gravity (LT -2 ).Te riction actor depends on eynolds number Corresponding Autor: Moammad N. BagatElNesr, Mailing Address: King Saud University POB 2460 iyad 11451, Kingdom o Saudi Arabia. d r n e s g m a i l. c o m ; Tel: Fax:

2 ( N ) and te relative rougness o te pipe ( ). could be evaluated i N and are known by grapical or analytical means, troug Moody diagram (Moody, 1944) or Curcill equation (Curcill, 1977) respectively. v D (2) N e D (3) were : water density (ML -3 ), v: velocity (LT -1 ), D: pipe inner diameter (L), : viscosity (ML -1 T -1 ), and e: mean rougness eigt along pipe inner surace. Altoug tere are several equations used to evaluate te riction actor, but te Curcill equation, Eqn.(4),is te only one valid or all types o low, laminar, turbulent, and even transient.for tat reason, Curcill equation is used in tis work ln 7 N 0.27 N (4) N As noticed, Curcill equation requires to be known, wile it is not easy to be measured on all pipe lietime. Moreover, e varies due to te quality o water used, quantity o sediments in it, age o pipe, pipe wall material and inising, and some oter minor causes.accordingly, most o te designers use Hazen-Williams equation, Equ. (5), wic depend only on a single actor called capacity actor (C)wic relies on te pipe material and age, were it varies rom 80 to 150 rom very roug to very smoot pipes. (Williams and Hazen, 1933) L Q D C WereK: units parameter =1.21E10 wen substituting Din [mm], Qin [l/s], and Lin [m]. In tis study, a simple iterative model were developed to establis terelationsip between Hazen Williams C and Darcy-Wiesbace/D, in order to ind te amount o rectiication needed to C to make te usage o H-W equation more accurate. Model Development: In designing a drip irrigation subunit, te allowable riction loss is adjusted to minimize variations between emitters in te subunit not to exceed 10% o te emitter s nominal discarge (ASABE, 2008).Tis amount o discarge tolerance is converted to pressure troug te emitter equation, Eqn.(6), so as it is aected by te emitter low exponent x, as sown in Eqn.(7). q dq q x k (6) d x (7) werek: emitter low parameter [L 3 T -1 ], : pressure ead [L]. As noticed by Eqn.(7), or an emitter wit x=0.5; 10% variation in discarge is equivalent to 20% variation in pressure ead.however, tis amount is usually distributed between laterals and maniolds as 55% and 45% respectively. For example, i te emitter s equation is q= , so as ten te allowable pressure variation is 20% o 10m, i.e. 2m. Tereore, te allowable losses in lateral lines and maniolds are 1.1m, and 0.9m respectively. Most o te designers allow te emitter to operate minimally on its nominal discarge, so te ar-end emitter in te subunit will operate at 10m ead, and te near-end emitter will operate at 12m ead, as illustrated in Fig. 1. Starting rom te ar-end emitter, wit te minimum desired discarge, moving against water direction, riction losses are calculated or eac segment depending on te passing low, wic increases in te oppositelow direction as illustrated in Fig 2. On eac segment, total riction losses are summated and compared to te allowable riction loss in laterals. Te suitable lateral line lengt is calculated as in Eqn(8). (5) 1080

3 i n i Ten : L ( n 1) (8) i 1 allowable in line Suitable i Consequently, te allowable line lengt varies according to te riction equation. D-W metod is considered as te proper reerence or allowable line lengt. Hence, te related H-W actor C could be establised. However, establising C cannot be done directly,as te riction loss o te lineis computed rom summation o segments losses.so, an iterative metod sould be used.for eac relative rougness value, te reerence allowable lengt (L all ) was ound using D-W metod ten, L all was establised again using H-W metod wit C values in te range 70 to 160.Te equivalent Cestimate is te value tat leads to te closer allowable line lengt to te D-W metod. For example, i D-W gives anl all equal to 52m, wile at H-W metod wit dierent values o C wit teir corresponding Lall values. An interval ound wit boundaries o L all. 50m and 60m atc=120, and 125 respectively.te closer Cvalue sould be 120. But or more accurate calculations, C sould be evaluated wit te relative interval metod,eqn.(9), in tis case C=121. C C1 LDW L1 C L C 2 1 L 2 1 (9) werel DW is te L all value establised rom D-W metod, subscripts 1 and 2 indicate te irst and second boundaries o te interval were L DW laid inside, as mentioned above. To establis te relationsips between C and te related variables; a spreadseet model was developed; wereas te eect o eac variable was cleared. Te variables and teir values are sown in Table 1. A detailed lowcart o te developed model is illustrated in te appendix. ESULTS AND DISCUSSION elative rougness (e/d) versus capacity actor (C): A simulation run was perormed or eigt values o pipelinesdiameter as mentioned in Table 1.Te irst tree diameters were considered as lateral lines, wile te rest ive diameters were considered maniolds. For eac case, te nine relative rougness values were applied and te corresponding capacity value was computed.te entire simulation was executed or te two mentioned nominal discarges o te emitter. Te simulation results are summarized in Fig 3. As noticed in Fig. 3, te C value tends to increase as te rougness decreases (8 is te rougest pipe and 0 is almost very smoot), tis coincides wit te typically expectedtrend.however, in H-W equation, te C value is set to be 150 or very smoot pipes, and te sown trends approaces tis value in most cases. Te sown diameters were classiied into two groups; laterals group and maniolds group. In te laterals group C value starts rom about 70 (at 8) to about 130 (at 0).Te 4L/ discarge emitter C values are almost 10 units less tan teircorresponding values o te 2 L/ emitter. On te oter and, te maniold group is uniorm, starting rom below 70 units to unite in te standard value o 150 (at 0).Tis results lead us to conclude tat using C=150 wit H-W equation in designing drip lateral lines isincorrect, wile using C in te range 130 to 135 is more reliable to get accurate results. Wile using C=150 or designing maniolds is acceptable or very smoot pipes like PVC, and or te early ages o te pipe beore rougness increases due to sediments and cemicals. However, i te designed network issupposed to suer lack o maintenance and management, ten te C actor sould lose 10% o its value to be about 135 or maniolds as a actor o saety. InFig. 4, it can be noticed tat emitter discarge eect is very small, wile te rougness series are divided into two groups; smoot group and roug group. Smoot group contains only one curve (0), wile te rest o relative rougness levels are set in te oter group. Te capacity actor tends to increase wit te pipe diameter (D) in te smoot group, wile it contrarily decreases wit D in te roug group.tis could be contributed tat or roug pipes, as D increaseste mean rougness e also increases (as te relative rougness is ixed). Tereore, te equivalent capacity actor moves toward te rougness direction (less C value). For te smoot group, as D increases te low resistance decreases (wit no wall rougness), tereore te equivalent capacity actor moves against te rougness direction (more C value). Line Lengt As Aected By Pipe ougness and Emitter Exponent: Determining line lengt is a main goal wile planning or designing drip networks. However, many designers consider te line lengt as an experience-mentioned property. Actually, line lengt varies widely wit te amount o low, emitters caracteristics, and pipe rougness. Fig. 5 sows a sample illustration o a 13 mm lateral line wit dierent combinations o emitter discarge, emitter low exponent, and pipeline rougness. It could be noticed tat line lengt is inversely proportional to emitter discarge, emitter exponent, and to pipe 1081

4 rougness as well. It is noticeable tat te line lengt resulted rom using H-W wit C=150 is longer tan tat od-w at 0. Tis is because te overestimate o C=150 to express te lateral lines as mentioned beore. However, designing lateral lines using H-W eqn. wit C=150 leads to longer lines, and tereore to drop in perormance and increase in riction losses. Te increment in riction losses was calculated by te model as ollows: FIP H. W D. W (10) DW. were FIP: riction increment percent due to using H.W instead o D.W, (%), : riction ead loss in line (L), H.W: calculated using Hazen-Williams metod, D.W: calculated using Darcy-Weisbac metod. a brie study o lateral diameters is illustrated in Table 2. In tis table, it is noticed tat FIP varies rom 6% to 18.4%, tis value increases wit te increment o x in most cases. It, owever, increases wit te decrease o emitter low rate, and decreases as te diameter increases. Tese results outsoot te importance o te proper assignment o te C value as mentioned beore. It could also be noticed in Fig. 5 tat te eect o emitter low-exponent (x) is very impressive, it may result to more tan 200% increase in te line lengt and in te 2 L/ cart). Tis leads to te importance o using pressure-compensating emitters or at least as low x values as possible. Te H-W coeicient C is directly aected by xtoo, tis could be attributed as te inconsistency o te low increases by te increase o x, so tat te riction inside te pipe increases wic leads to aigerc value as sown in Table 3. C Values Deviation From No-Exits Laterals: Te current model deals only wit lateral lines wit emitters installed, wile all o te mentioned literature deal wit no-exits lines. Fig. 6 illustrates te resultedranges o te current model, compared to te results o two o te no-exits works. It can be noticed tat te current model s range o 16mm pipeline lies witin te range o te two oter models, wile it as some bias below average in te 19mm pipeline, and above average in te 13mm pipeline. Tis bias could be attributed to eect o te emitters existence, were te line as a continuous decreasing lowrate. It could also be noticed rom Fig. 6 tat te proposed model is less spreading in value tan te oter odels, so an average value o C could be taken saely wit minimum error. Table 4 sows te standard deviation comparison o eac researc work.equation (11)sows a simple relation to obtain Cas a unction o x, D, and q. C D q x (11) Teadjusted correlation coeicient o te equation isr adj = and te standard error is, SE: Summary and Conclusions: Owing to te importance o Hazen-Williams equation in designing drip irrigation networks, te riction actor o it was adjusted to give te same results as te accurate Darcy-Weisbac equation. To acieve tis goal, a simple iterative model was developed, a comparative analysis was made, and a simple equation was presented to compute te appropriate C values. Te results o tis paper agrees wit previous works wit some bias due to dierence in analysis metods, and because tese works dealt wit a no-exits lateral line wile tis paper dealt wit real state laterals wit emitters installed on it. Te results sowed tat using te proper values reduces te riction loss error by up to 18.4%, and ence, lead to saer and more reliable drip irrigation networks. Table 1: Variables used in te model and its values. Values Variable From To Step Count x C (e/d) Label Value q (nominal) (2 l/), (4 l/) 2 D (13mm), (16mm), (19mm), (50mm), (63mm), (75mm), (90mm), (110mm) 8 Number o alternatives

5 Symbol C D D -W e ( ) F IP g a ll e m, H [L] Meaning water density [ML -3 ] viscosity [ML -1 T -1 ] Hazen-Williamscapacity actor pipe diameter [L] Darcy-Weisbac mean rougness eigt along pipe inner surace Darcy-Weisbac riction actor unction o riction increment percent acceleration o gravity [LT -2 ] emitter pressure ead [L] allowable ead loss [L] emitter ead [L] pressure ead loss due to riction [L] minimum allowable ead in te subunit [L] Nomenclature Symbol H -W i, j, k K k l L L all Q q, q em q s N v x Meaning Hazen-Williams Counters (in lowcart) units parameter emitter low parameter [L 3 T -1 ] pipe/segment lengt [L] Total pipe lengt [L] Allowable lengt [L] pipe discarge [L 3 T -1 ] emitter discarge [L 3 T -1 ] segment discarge [L 3 T -1 ] eynolds number relative rougness o te pipe water velocity [LT -1 ] emitter low exponent.. sould be increased by min Fig. 1: Drip irrigation subunit, sowing extreme emitters discarge and operating pressure Table 2: Maximum lengt o lateral lines wit dierent diameters calculated using bot D-W ormula (=0), and H-W ormula (C= =150). Friction Increase Percent (FIP) and Equivalent C at 0 are sown. 2 L/ 4 L/ x L DW-0 L HW-150 FIP Equv.C x L DW-0 L HW W-150 FIP Equv.C % % % % % % % % % % x L DW-0 L HW-150 FIP Equv.C x L DW-0 L HW W-150 FIP Equv.C % % % % % % % % % % x L DW-0 L HW-150 FIP Equv.C x L DW-0 L HW W-150 FIP Equv.C % % % % % % % % % % mm 16 mm 19 mm

6 Table 3: Hazen-Williams capacity variable (C) calculated by comparing allowable lateral lengts to Darcy-Wiesbac values or several emitter low-exponents. Lateral line* allowable lengt wen calculating riction losses using: Proper C Hazen-Williams Equation wit C value= Value Emitter Darcy- C low exponent 80 Wiesbac Equation Value * Tis case: D=13 mm, e=0.5 mm, q=4 L/ Average C or tis case: Fig. 2: Segment by segment calculation o drip laterall line. Table 4: Average and Standardd Deviation values comparison o H-W C actor. q em D Proposed Mogazi (1998) Yildrim and Ozger (2008) L/ Mm Avg SD Avg SD Avg SD

7 Fig. 3: elative rougness versus capacity actor, compared at several pipeline diameters and two emitter nominal discarges. Fig. 4: Pipeline diameters eect on capacity actor, at several relative rougness values and two emitter nominal Discarges.

8 Fig. 5: Maximum allowable line lengt o a 13mm lateral line wit dierent emitter low exponents, emitter discarges o 2 and 4 L/, or several pipe rougness mm 4 L/ 16 mm 13 mm Proposed Model Mogazi (1998) Yildrim and Ozger (2008) 19 mm 2 L/ 16 mm 13 mm H-W C values ranges Fig. 6: Comparison cart between C ranges resulted rom current model and two oter models. 1086

9 Fig. 7: Flowcart representing te model procedure or determining te equivalent riction actors o H-W and D-W. (Symbols are deined in te nomenclature). ACKNOWLEDGEMENT Te autors wis to express teir deep tanks and gratitude to Saik Moammad Bin Husain Alamoudi or is kind inancial support to te researc cair Saik Moammad Alamoudi Cair or Water esearces (AWC), ttp://awc.ksu.edu.sa, were tis study is part o te AWC activities in te Projects & esearc axis. EFEENCES Allen,.G., elating te Hazen-Williams and Darcy-Weisbacriction loss equations or pressurized irrigation, Applied Eng. in Agric., 12(6): ASABE, Design and Installation o Microirrigation Systems, ASAE EP405.1, AP1988 (2008).In ASABE Standards ASAE: St. Josep, MI, 5pp. UL: ttp://asae.rymulti.com/standards.asp, accessed Nov Curcill, S.W., Friction actor equation spans all luid low regimes. Am. Inst. Cem. Eng. J. 23: Martorano, S., Calculatingriction loss: Darcy-Weisbacormula vs. Hazen-Williams, Tecnical Article. Viking Corp. pp: 8., UL(accessed Oct. 2008): ttp:// Mogazi, H.M., Estimating Hazen Williams Coeicient or Polyetyline Pipes. J. Transp. Eng. 124(2): Moody, L.F., Friction actors or pipe low. Trans. ASME, Quoted rom Allen, 1996., 66(11): Sayyaa, W.H. and S.S. Sablani, An artiicial neural network or non-iterative calculation o te riction actor in pipeline low.computers and Electronics in Agriculture., 21(3):

10 Valiantzas, J.D., Modiied Hazen Williams and Darcy Weisbac Equations or Friction and Local Head Losses along Irrigation Laterals.J.Irrig. Drain. Eng., 131(4): Williams, G.S. and A. Hazen, Hydraulic Tables, 3 rd Ed., ev. New York, N. Y.: Jon Wiley & Sons, Inc. Yildirim, G. and M. Ozger, Neuro-uzzy approac in estimating Hazen-Williams riction coeicient or small-diameter polyetylene pipes, Adv. Eng. Sotw. Doi: /j.advengsot (Article in press). 1088