EVALUATION OF THE CHRISTIANSEN METHOD FOR CALCULATION OF FRICTION HEAD LOSS IN HORIZONTAL SPRINKLER LATERALS: EFFECT OF VARIABLE OUTFLOW IN OUTLETS *

Transcription

1 IJST, Transatons o Cvl Engneerng, Vol. 35, No. C, pp Prnted n Te Isla Republ o Iran, 0 Sraz Unversty EVALUATION OF TE CRISTIANSEN METOD FOR CALCULATION OF FRICTION EAD LOSS IN ORIZONTAL SPRINKLER LATERALS: EFFECT OF VARIABLE OUTFLOW IN OUTLETS * S.. SADEGI, S. F. MOUSAVI ** M. GEYSARI AND S.. R. SADEGI Water Eng. Dept., Isaan Unversty o Tenology, Isaan , I. R. o Iran Eal: ousav_s@yaoo.o College o Natural Resoures and Marne Senes, Tarbat Modares Unversty, Noor , Mazandaran, I. R. o Iran Abstrat Te Crstansen's rton ator, F, s usually used to opute ead loss due to rton n laterals o pressurzed rrgaton systes. In ts resear, ts ator was evaluated onsderng varable dsarge or outlets loated along te laterals. For ts purpose, two geoetral progressons were establsed between te outlets dsarges. Te rst progresson was apponted between te dsarge o te rst outlet and te outlet w s assued to possess te average pressure. Te next progresson was developed between te dsarge o ts outlet and te ost downstrea outlet w s assued to operate at least dsarge. Te proposed F-ator s a unton o nuber o outlets, te pertted varaton o pressure between te rst and te last outlet, and also te orula beng used to alulate rton ead loss along te lateral. Te results sowed tat Crstansen s etod aurately estates total rton ead loss along te laterals su tat axu relatve error aused by assung equal dsarges n outlets does not exeed 3%. Keywords Frton ator, ead loss, geoetral progresson, lateral ppelne. INTRODUCTION In a pressured rrgaton syste or any oter ydraul syste nludng ultple outlets along te ppes, estaton o te total rton ead loss along a lateral requres a stepwse analyss startng ro te lowest outlet, workng upward and oputng te ead loss aused by rton n ea segent. Nowadays, te nreasng progress n oputer tenology as led to te developent o nueral etods [- 4] w allow easy evaluaton o ead losses along te lateral. Tese odels, owever, ave a g level o sopstaton and requre extensve prograng and, peraps, long oputer-exeuton te [5]. In at, altoug te stepwse oputaton o te rton loss as been greatly asssted by te use o spreadseets, orreton ators ontnue to be used n te lterature [6, 7]. Several researers ave developed te onept o te reduton ator or losed-end ultple-outlets ppes, begnnng wt Duput [8], or ontnuous outlow water ppes, and ollowed by Crstansen [9] or dsrete outlow rrgaton lnes onsderng te ollowng assuptons: No outlow exsts at te downstrea end o te ppelne (losed-end ppe. All outlets are equally spaed (onstant outlet spang. 3 All outlets ave equal dsarge, and 4 Te dstane between te ppe nlet and te rst outlet s equal to one outlet spang. Reeved by te edtors Mar 3, 00; Aepted Deeber 6, 00. Correspondng autor

2 34 S.. Sadeg et al. Oter researers ave attepted to aount or te eet o te rst outlet s spang lengt upon tat ator sne Jensen and Fratn [0] developed a soluton or dsrete outlow wt te rst spang lengt equal to al o te ull outlet spang. Cu [] developed a slgt odaton to te onepts o te prevous researers, wle Salopp [] generalzed te orulaton wt a relatve soluton tat uses te Crstansen [9] ator as a base or alulaton. Cuena [3] presented a soluton to te Jensen and Fratn [0] ondton wle DeTar [4], Valantzas [5], and Alazba [6] arrved at results slar to Crstansen [9]. An alternatve treatent n te alulaton o total rton loss onsders te eet o outlow varaton n alulaton o te F ator [5, 5, 7-9]. Tese approaes, owever, are based on ntrodung an equaton or te F ator or unor outlow along te lateral and ten adustng t by a ore onrete oason n w te outlow s onsdered to be non-unor. Neverteless, any o tese studes ave not speed te aount o error aused by assung equal dsarges or outlets n oputaton o te F ator. Tereore, te obetve o ts study was to evaluate te Crstansen s [9] etod or alulaton o total rton loss or a orzontal sprnkler lateral by onsderng derent dsarges n outlets and also to deterne te axu error aused by ts proedure. In oter words, we ntend to dsss te onventonal assupton n w total low at te nlet s equal to N q ( q s dsarge o te outlet w possesses te average ead, and N s total nuber o outlets. Ts goal s ollowed by assung two logal geoetral progressons along te lateral. It s to be noted tat n a ppelne wt ultple outlets, tere wll be energy losses aused by te ouplngs and te struture o te outlet usually alled loal losses. owever, tere s also a gradual reduton n te veloty ead as low passes te outlets and ts wll ause an nrease n pressure, w wll balane losses aused by turbulene at outlet ouplngs []. ene, an exat proedure to alulate pressure losses n ppelnes wt ultple outlets annot be usted [0]. Tese assuptons also underlne te present work. a Deternaton o geoetral progressons. ANALYSIS Consder a orzontal ppelne nludng N outlets (Fg.. On ts ppe, te rst and last outlets operate wt axu and nu dsarge, respetvely. Te outlet w works wt an average ead s loated at approxately 40% o te lateral lengt ro te nlet [7, 7, ]. So, te nuber o outlets (N on te lateral ppelne s a ultple o 5, te outlet untonng wt an average ead wll be te 0.4Nt. Introdung te average orreton ator, F AVG, Anwar [] reports tat (-F AVG o te total rton ead loss ours beore te outlet avng an average pressure ead. Te value o (-F AVG s sown n Fg.. It s evdent tat perent o total rton ead loss (N to N500, respetvely ours beore te outlet avng an average pressure ead. So ore dsarge varaton s expeted along te ppe aead o te outlet w works wt te average ead tan ater t (.e., te reanng 60%. In oter words, te rate o te outlets dsarge varaton beore te outlet wt te average ead wll not be equal to te rate o dsarge varaton or te rest o te lateral. eurstally, n order to onsder varable dsarges or outlets and also oer ore lexblty to te odel, we deded to onsder two geoetral progressons. Te rst s apponted between te rst outlet s dsarge, loated at te upper part o te ppe, and te dsarge o te outlet w operates wt average ead (te 0.4Nt outlet. Te seond progresson s developed between te 0.4Nt outlet and te last outlet on te lateral. It s assued tat te rst outlet were postoned at te nlet o te lateral, ts dsarge would be q but ts dsarge anges to or te outlet at a dstane l ro te nlet, were s te rst progresson value ( <. Mateatally, te dsarge o te seond outlet redues to and te 0.4Nt outlet s IJST, Transatons o Cvl Engneerng, Volue 35, Nuber C August 0

3 Evaluaton o te Crstansen etod or alulaton o 35 dsarge ust be, were 0.4N. Consderng te paraeter or te seond progresson ( <, te last outlet s dsarge wll be, were 0.6N. Te varaton o dsarge o te outlets s sown seatally n Fg.. Fg.. Varable dsarge or N outlets along a lateral. Te rst outlet s one ull- spang ro te nlet Fg.. Te value o (-F AVG or derent values o N and [] To alulate and, tree logal assuptons ave been ade: - To antan a unorty oeent o about 97%, total pressure varaton (ead loss n a lateral ppelne wt nuerous outlets s typally lted to 0% [7]. Ts quantty was denoted as P. ene we ave: Δq ΔP ( were q and P are total dsarge varaton and total pressure varaton along te lateral, respetvely. For nstane, te total ead loss pertted along te lateral s 0%, ten P 0. and q For a orzontal lateral, te outlet w operates wt average pressure ead s loated approxately n 0.4L, and tereore ts dsarge s. 3- Aordng to Fg. 3, t s assued tat λ ours between te rst outlet and te outlet w operates wt average pressure ead, and tereore (-λ ours ater ts outlet. Note tat te value o λ wll be oputed n seton (. Fg. 3. Deternaton o paraeter λ August 0 IJST, Transatons o Cvl Engneerng, Volue 35, Nuber C

4 36 be S.. Sadeg et al. An equvalent ter to Eq. ( s tat te rato o te rst outlet's dsarge to te last one an ΔP. ene we ave: ΔP ( Consderngδ ΔP, Eq. ( an be wrtten as sown below: (3 δ In a pressurzed sprnkler rrgaton syste, dsarge o an outlet on a lateral s a unton o pressure ead, and an be wrtten as: k 0.5 q (4 were q s te outlet s dsarge, s ead at te outlet, and k s a onstant oeent w depends on te outlet aratersts. Applyng Eq. (4 or te rst outlet, te outlet w operates wt average pressure, and te last outlet, we wll ave: (5 k k (6 N k were s ead at te rst outlet, s ead at te outlet workng wt average pressure, and N s ead at te last outlet. Consderng te trd above-entoned assupton and also Eqs. (5 to (7, we an wrte: k ( k ( (8 λ ( ( ( λ (9 k k were λ s a onstant value desrbng te total rton ead loss beore te outlet w operates wt average pressure ead. Substtutng Eq. (3 nto Eq. (9 and rearrangng gves: q ( ( ( λ (0 k δ Also, splyng Eq. (8 yelds: q ( ( λ ( k (7 IJST, Transatons o Cvl Engneerng, Volue 35, Nuber C August 0

5 Evaluaton o te Crstansen etod or alulaton o 37 Paraeter s ound by dvdng Eq. ( by (0: λ λ ( δ ΔP n Eq. ( and A better or or te rst progresson value ould be ound by substtutng δ by rearrangng, w yelds: ΔP( λ (3 ΔP On te oter and, splyng Eq. (3 gves te paraeter : b A new approa to develop F ator (4 ΔP Crstansen [9] proposed te F ator or alulatng total rton ead loss n a lateral ppelne wt ultple outlets. We tred to relate our analyss wt ts ator and nd an analyt relaton or t. Te general equaton or oputng rton loss n a ppe an be sown as: TD were s total rton ead loss, T s a oeent based on te rton orula used, D s ppe daeter, L s lengt o te lateral, s total dsarge, and n and are exponents. Consderng ρ TD n to be a onstant ter, te rton ead loss or ea segent ( llustrated n Fg. ould γ be wrtten as (γ represents te segent nuber: ρl ρ l( ρ l( 3.. ρl(... n L ρl(... ρl(... ( ρl(. N ( ρ l(... (6 Fro Eq. (6 we an wrte: (5 August 0 IJST, Transatons o Cvl Engneerng, Volue 35, Nuber C

6 S.. Sadeg et al. IJST, Transatons o Cvl Engneerng, Volue 35, Nuber C August ( % ρ l (7 0 ( (... 60% ρ l N N (8 were 40% s te total rton ead loss n 40% o te upper seton o te lateral and 60% s te total rton ead loss n 60% o te lower part o te lateral. ene: 60% % 40 (9 Substtutng Eq.(7 and (8 nto Eq.(9 leads to: 0 0 ( ( ( ρ l (0 Total dsarge o te lateral s te su o all outlets dsarge. Tereore: ( ( ( Substtutng Eq. ( or n Eq. (0 gves: 0 0 ( ( ρ lq ( Consderng Eq. (5 and LNl we ave: n FTNlD (3 By substtutng Eq. ( n Eq. (3 we ave: Nlq F ( ( ρ (4 Fnally, Eqs. ( and (4 are equated, ten: N F ( ( 0 0 ( ( (5 Note tat te proposed F ator ere s ndretly related to te pressure ead varaton along te lateral beause and are related to ts paraeter.

7 Evaluaton o te Crstansen etod or alulaton o 39 Table sows a oparson between te F values alulated ro Eq. (5 and Crstansen's F ator or.85 and derent values o P. It s seen ro ts table tat, as P nreases, te value o te proposed F ator dereases. Also te Crstansen s F ator s very lose to F o Eq. (5 wen P0.05 sne, or ts ase, te values o te geoetral progressons are lose to.0 and te assupton o equal dsarges n outlets s ore logal. Fnally, t s onluded tat a axu relatve error o.% ay our or N65 and P0.. Deternaton o λ Table. F-ator or.85 and derent values o P Nuber F (alulated by Eq. (5 o outlets ΔP 0. ΔP 0. 5 ΔP 0. ΔP F C In order to alulate te value o bot geoetral progressons (.e., Eqs. (3 and (4, te value o λ ust be derved. For ts purpose, onsder a orzontal ppelne as llustrated n Fg.. Paraeters l,, and are sown on te gure. Te ollowng relatonsp an be wrtten or te pressure at te ppe nlet, l, and te pressure ead o te rst outlow, : l ρ l (6 On te oter and, te relatonsp between te nlet pressure and te outlet, w operates at average pressure or zero ground slopes s [7]: Consderng Eqs. (3, (6 and (7, Eq. (7 an be transored to: l 3 ( NF (8 August 0 IJST, Transatons o Cvl Engneerng, Volue 35, Nuber C

8 40 S.. Sadeg et al. were F s te Crstansen's F ator. Coparson o Eq. (8 and our trd assupton sows tat te 3 te ( n Eq. (8 s equal to λ. Te equaton or dretly alulatng te F ator s gven by 4 NF DeTar [4] as: F N (9 N Tereore, by substtutng Eq. (9 n (8, λ s ound as: 3 λ (30 4 N 0.5 N Table sows te value o λ or derent nuber o outlets and values. As te nuber o outlets nreases, λ approaes 0.75, w s proposed by Keller and Blesner [7]. owever, te observed derene between tese two oeents s beause λ represents te rton ead loss between te rst outlet and te outlet w operates under average pressure, wereas te oeent 0.75 onsders te rton loss between te nlet o te lateral and te outlet wt average pressure ead. Nuber o outlets Table. Values o λ or derent values o and N λ Te alulated λ ere oers a new or o alulatng w ould be derved by substtutng Eq. (30 n Eq. (3: IJST, Transatons o Cvl Engneerng, Volue 35, Nuber C August 0

9 Evaluaton o te Crstansen etod or alulaton o 4 ΔP 0.5ΔP N 0.5 N Δ (3 P and are alulated ro Eqs. (3 and (4 and are suarzed n Table 3 or derent nuber o outlets along te lateral, our values o P, and.85. As t s seen n te table, bot geoetral progressons approa.0 wen te nuber o outlets nreases. Also, t s evdent tat as P nreases, te value o and dereases. Nuber o outlets Table 3. Varaton o and or.85 and derent values o N and Ps ΔP 0. ΔP 0. 5 ΔP 0. ΔP Te oter pont ere s tat aordng to Table 3, s always less tan. Logally ts eans tat or a orzontal lateral, te rate o te outlets dsarge varaton s ore severe beore te outlet possesses te average pressure ead. d Total dsarge o te lateral Te onsderaton o two geoetral progressons not only aets te F ator, but wll also nluene te total low enterng te lateral nlet sne s equal to te suaton o all outlets dsarge. In order to approve ts eet, Eq. ( s rewrtten as: ( ( ( (3 August 0 IJST, Transatons o Cvl Engneerng, Volue 35, Nuber C

10 4 S.. Sadeg et al. On te oter and, based on our seond assupton, (3 takes te ollowng or: q ( ( s approxately equal to q. Tereore, Eq. ( In order to opare te oputed total nlow by te suggested odel and te onventonal approa by Crstansen (94 n w Nq, a seres o nueral oputatons were arred out. For ts purpose, a obnaton o nput paraeters N, and P were onsdered. Te value o N was vared between 5 to 00 by an nreent o 5; was osen as.75,.85,.9 and ; and our values were onsdered or P as P 0., 0.5, 0. and Usng tese data, te values o and were alulated usng Eqs. (3 and (4. Te value o / q was obtaned usng Eq. (33. Te relatve error between ts quantty and te value o total nlow at te nlet ( Nq was oputed and plotted aganst te total nuber o outlets n Fg. 4. Ts gure deonstrates tat relatve errors are neglgble and are always between -0.5% and %. (33 Fg. 4. Relatve errors n alulaton o e Deternaton o Δ P / q usng Eq. (33 and te onventonal approa ( Nq To deterne P, we onsder a orzontal lateral ppelne on w N outlets, wt a spang o l and equal dsarge o q, are loated. Negletng te veloty ead along te lateral, w s aeptable or drp and sprnkler rrgaton systes [], we an wrte: On te bass o our rst assupton, we ave: Equatng Eqs. (34 and (35 gves: Consderng Eqs. (4 and (3: (34 l N l N ΔP (35 Δ P (36 a IJST, Transatons o Cvl Engneerng, Volue 35, Nuber C August 0

11 Evaluaton o te Crstansen etod or alulaton o 43 Fnally, usng Eq. (9 yelds: ρk N L F Δ P (37 N N ΔP ρ k L ( (38 It s observed tat P depends on te nuber o outlets, ppe lengt and type, total dsarge o te ppe, oeent, and ppe daeter. 3. COMPARISON TO TE CRISTIANSEN S METOD An exaple s gven ere to llustrate te applaton o te adusted rton oeent (Eq. 5 and alulaton o total rton ead loss along a lateral. Suppose tat a weel-ove sprnkle rrgaton syste s desgned su tat 5 outlets are xed on an alunu ppelne. Te spang between outlets and also between te rst outlet and te lateral nlet s 0. Calulate te total rton ead loss or ts ppe by usng bot Crstansen s F-ator and te etod proposed n ts paper. Suppose tat te daeter o te ppe s 3, te desgn dsarge o te outlets s 0.9 lter/se wt an average pressure ead equal to 35. Te rton loss s assued to be desrbed by te azen Wllas orula [C W 30, or alunu ppes wt ouplers approxately every 0 [7]]. Soluton. Crstansen etod: For.85 and 5 outlets, F C (Table. Te total low at te nlet o te lateral s alulated as Nq5 0.9lps 3.5 lps. Te azen-wllas orula or alulatng rton ead loss n te SI syste s: D 0.67 F L( CW were C W s te azen-wllas oeent FL( D (.85( CW (39. Te etod o ts paper: - Usng Eq. (38, P Fro Table 3, or P0. and N5, and Fro Table, te adusted F-ator or 5 outlets (or.85 and P0. s ; q lps 5- Total dsarge o te lateral (ro Eq. ( s: ( ( lps August 0 IJST, Transatons o Cvl Engneerng, Volue 35, Nuber C

12 44 S.. Sadeg et al FL( C W ( D ( IJST, Transatons o Cvl Engneerng, Volue 35, Nuber C August Ts exaple deonstrates tat te Crstansen [9] etod slgtly overestates te total rton ead loss along te lateral (te relatve error ere s.6%. Fabratng a test rg n order to evaluate te Crstansen s F ator n anolds, Moaed et al. [3] also reported tat te value o F C was relatvely greater tan te experental F ator. owever, Salopp and Allen [] onsdered errors o su an order o agntude to be wtn te range o auray o pressure gauges oonly used n sprnkler rrgaton systes. It s wort notng tat tese errors wll sgnantly derease te nuber o outlets along te ppe nreases. A reent study by Yldr [8] also testes to ts stateent. Solvng an exaple or te ydraul desgn o a sprnkler rrgaton syste, Yldr [8] reports tat te Crstansen oeent or unequal dsarges o te outlets and.85, l, q 0.35 L/s and N 33 was alulated as 0.367, wle Crstansen [9] gave F or N 33. Our etod also gves F 0.36, w sows tat te assupton o onsderng equal dsarge n te outlets beoes logal wen te total nuber o outlets s large. 4. CONCLUSION In ost sprnkler rrgaton systes, te outlets are unorly spaed along te lateral ppelne. Beause o te portane o estatng te total rton ead loss along te lateral, Crstansen [9] proposed te rton oeent ator F and subtted a table or s results. In te present resear, ts etod was evaluated by onsderng varable dsarge n outlets n order to quanty te relatve error aused by assung equal dsarge n outlets. For ts purpose, two geoetral progressons were assued to exst between te outlets dsarges. Results sowed tat te value o te F ator anges as te total pressure ead varaton along te lateral anges, and altoug Crstansen s etod and our proposed equatons gve approxately slar values or F ator, Crstansen [9] slgtly overestates te total rton ead loss along te lateral, partularly wen N 0 and P 0.. owever, ts overestaton an be negleted sne te axu relatve error between te two etods does not exeed 3%. REFERENCES. asanl, A. M. & Dandy, G. C. (000. Applaton o genet algorts or optzaton o drp rrgaton. Iranan Journal o Sene and Tenology, Transaton B: Engneerng, Vol. 4, No., Mousav, S. F. & Nour-Eazade, M. R. (00. Applaton o nueral etods n water resoures. Arkan Publ. Co., Isaan, 38 p. 3. atoot,. M., Abo-Gobar,. M., Al-Aoud, A. I. & Moaad, F. S. (994. Analyss and desgn o sprnkler rrgaton laterals. J. Irrg. and Dran. Eng., ASCE, Vol. 0, No. 3, pp Kang, Y. & Nsyaa, S. (996. Analyss and desgn o rorrgaton laterals. J. Irrg. and Dran. Eng., ASCE, Vol., No., pp Valantzas, J. D. (998. Analytal approa or dret drp lateral ydraul alulaton. J. Irrg. and Dran. Eng., ASCE, Vol. 4, No. 6, pp Jaes, L. G. (988. Prnples o ar rrgaton syste desgn. Jon Wley and Sons, New York, NY, 543 p. 7. Keller, J., & Blesner, R. D. (990. Sprnkler and trkle rrgaton. Capan & all, New York. 8. Duput, J. (865. Traté téorque et pratque de la ondute et de la dstrbuton des eaux, Duond, ed. Ibare des Corps Ipéraux, Frane, pp

13 Evaluaton o te Crstansen etod or alulaton o Crstansen, J. E. (94. Irrgaton by sprnklng. Cal. Agr. Expt. Sta. Bull. No. 670, Unversty o Calorna, Davs, Cal. 0. Jensen, M. C. & Fratn, A. M. (957. Adusted F ators or sprnkler lateral desgn. Agr. Eng., Vol. 38, No. 4, p Cu, S. T. (978. Moded F ator or rrgaton laterals. Trans. ASAE, Vol., No., pp Salopp, E. J. (988. Adusted F ator or ultple-outlet ppes. J. Irrg. and Dran. Eng., ASCE, Vol. 4, No., pp Cuena, R.. (989. Irrgaton syste desgn: An engneerng approa. Prente-all, Englewood Cls, N.J., 87 p. 4. DeTar, W. R. (98. Moded grapal deternaton o suban sze. Trans. ASAE, Vol. 5, No. 3, pp Valantzas, J. D. (00. Contnuous outlow varaton along rrgaton laterals: Eet o te nuber o outlets. J. Irrg. and Dran. Eng., ASCE, Vol. 8, No., pp Alazba, A. A. (005. Calulatng F ator or enter-pvots usng spled orula and oded Crstansen equaton. J. o Kng Saud Unversty. Agr. S., Vol. 7, No., pp Yldr, G. (007a. An assessent o ydraul desgn o trkle laterals onsderng eet o nor losses. Irrg. and Dran., Vol. 56, pp Yldr, G. (007b. Analytal relatonsps or desgnng ultple outlets ppelnes. J. Irrg. and Dran. Eng., ASCE, Vol. 33, No., pp Yldr, G. (00. Total energy loss assessent or trkle lateral lnes equpped wt ntegrated n-lne and on-lne etters. Irrg. S., Vol. 8, No., pp Par, C.., nz, W. W., Red, C. & Frost, K. R. (975. Irrgaton. 5t Ed., Irrgaton Assoaton, Farax, Va.. Salopp, E. J. & Allen, R. G. (993. ydrauls o rrgaton laterals: Coparatve analyss. J. Irrg. and Dran. Eng., ASCE, Vol. 9, No., pp Anwar, A. A. (000. Inlet pressure or tapered orzontal laterals. J. Irrg. and Dran. Eng., ASCE, Vol. 6, No., pp Moaed, T. A., Gazal, A.., Megat Mod Noor, M. J. & Mod Soo, M. A. (006. Deternaton and valdaton o G-ator or plast anolds. Int. J. Eng. and Tenol., Vol. 3, No., pp August 0 IJST, Transatons o Cvl Engneerng, Volue 35, Nuber C