Sprinkle and Trickle Irrigation
Sprinkle and Trickle Irrigation BIE 6110
Utah State University
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This 12 page Class Notes was uploaded by Mr. Tessie Labadie on Wednesday October 28, 2015. The Class Notes belongs to BIE 6110 at Utah State University taught by Gary Merkley in Fall. Since its upload, it has received 21 views. For similar materials see /class/230407/bie-6110-utah-state-university in Biology at Utah State University.
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Date Created: 10/28/15
Lecture 22 Numerical Solution for Manifold Locationl I Introduction In the previous lecture it was seen how the optimal manifold location can be determined semigraphically using a set of nondimensional curves for the uphill and downhill laterals This location can also be determined numerically In the following equations are developed to solve for the unknown length of the uphill lateral Xu without resorting to a graphical solution Definition of Minimum Lateral Head 0 In the uphill lateral the minimum head is at the closed end ofthe lateral furthest uphill location in the subunit 0 This minimum head is equal to hn39 hl hfu XuS where hn is the minimum head m h is the lateral inlet head m hfu is the total friction loss in the uphill lateral m Xu is the length ofthe uphill lateral m and S is the slope ofthe ground surface mm 0 Note that 8 must be a positive value hfd1 o In the downhill lateral the minimum head may be anywhere from the inlet to the outlet depending on the lateral hydraulics and the ground slope o The minimum head in the downhill lateral is equal to hn I hl 1 2 XmS Sprinkle amp Trickle Irrigation Lectures Page 241 Merkley amp Allen where hfu1 is the total friction loss in the downhill lateral m hfu2 is the friction loss from the closed end of the downhill lateral to the location of minimum head m and Xm is the distance from the manifold lateral inlet to the location of minimum head in the downhill lateral m o Combining Eqs 1 and 2 hfu SXu Xm hfd1hfd2 0 396 Ill Location of Minimum Head in Downhill Lateral o The location of minimum head is where the slope of the ground surface 8 equals the friction loss gradient J S J 397 where both 8 and J are in mm and Sis positive you can take the absolute value of S 0 Using the HazenWilliams equation the friction loss gradient in the downhill lateral at the location where S J is 1852 Ju Sefe qa Xu Xm D7487 398 Se 3600390 where J is the friction loss gradient mm Se is the emitter spacing on the laterals m fe is the equivalent lateral length for emitter head loss m qa is the nominal emitter discharge lph L is the sum ofthe lengths of the uphill and downhill laterals m Xu is the length ofthe uphill lateral m Xm is the distance from the manifold to the location of minimum head In the downhill lateral m C is approximately equal to 150 for plastic pipe and D is the lateral inside diameter mm o The value of 3600 is to convert qa units from lph to lps 0 Note that Xd L Xu where Xd is the length of the downhill lateral 0 Note that qaL Xu Xm3600 Se is the ow rate in the lateral in lps at the location ofminimum head Xm meters downhill from the manifold o Combining the above two equations and solving for Xm Merkley amp Allen Page 242 Sprinkle amp Trickle Irrigation Lectures 399 001293eCDZ63 see 03954 qa Sefe szL xu where the permissible values of Xm are 0 s Xm s Xd 0 Combine Eqs 396 amp 399 and solve for Xu by iteration 0 Alternatively based on Eq 87a from the textbook Xm can be de ned as 0571 3600 Se 50475 se Xm L Xu 5 400 qa 78910 Se fe IV Definition of Head Loss Gradients o In the uphill lateral the head loss is hfu Ju 39Fuxu 401 o In the downhill lateral the head losses are hfd1 Jd139Fd139 Xu 402 and hfd2 Jd2 39Fdz 39 Xu Xm 403 The above three F values are as de ned by Eq 89 in the textbook The friction loss gradients in mm are Ju39 K J x1 39852 404 Jd1 KJ L Xu1852 405 sz39 KJ L xu xm139852 406 where for the HazenV lliams equation Sprinkle amp Trickle Irrigation Lectures Page 243 Merkley amp Allen KJssf121210mD 487 e 407 V Solving for Optimal Manifold Location Usingtne oeti rlltlorls above Solve for the length of he uphlll lateral gtltu e gtlto Xu Notethatyou mignt oreter to use tne Darcerelsbach and Blasius equations tortne manitolo calculations they may be more accurate tnan Hazerlr Williams Tne ootManitolo computer program uses tne Darcerelabach amp Blasius equations Trickle Manifold Location ma Emlllel dlschalge llonl Lalelal lenoin lml 3 7m sun nun Emlllel Svaclng lml Lalelal lD lmml a 3 mm i7 EDD 11 glass Emlllel neoo lml Elmmd slave lmml ii Sun n m nun aoio lass le lml n nsn Res ulls tenoin oi oomll loieiol tenoin oi oownmll loieiol mo iiolo io minimum neoo MelkleyStAllen Paoem Spllnkle StTllckle lvvlgallun Lectures here do these E at ons Come From I Derivation of Nondimensionai Friction Loss Curves Tne nondimensionai friction ioss curves are iWim tne io actuaiiy one curve Wer nait iateraiiy inverted and shown as a dasned iine Fig 8 2 Tne dasned iine is Simin for ow in tne opposite direction Wnicn for our purposes is o 3 e previous iectures and from intuition tnattne upniii segment of iaterai pipe WiH not be more tnan Vathe iengtn because it is equai to 12 for tne case wnere tne ground siope is zero 5 g FoHoWirig is tne derivation for Eq 8 iob from Wnicn Fig 8 ZWas piotted Darcereiabach equation for circuiar pipes 408 Biasius equation or estimatingttor smaii diameter D lt i25 mm smootn pipes e PE amp PVC and based on more compiete equations tnat are used to piot tne Moody diagram 2032D25 409 wnere NR is tne Reynoids number wnicn for circuiar pipes is NR 9 We v er Tne kinematic Viscosityi vi is equai to about i 003it3 3 m25 forwater at Theri tortnis kinematic ViSCOSi i an 25 an 25 40 Q 2032 e00095 41 MD D Page 245 Meikiey StAHeri Putting the above into the DarcyWeisbach equation 7025 2 hf 00095 9 EV 412 D D 2g or h 0 00079LQ39 413 f W where hf is in m L is in m Q is in m3s and D is in m Eq 87a is obtained by having Q in lps and D in mm whereby the above coef cient changes to 7910 Finally in the above use LXL instead of L and QXL instead on and call it hm hfx 000079 UNUM 414 D 0 Then XLxL13975 xL23975 415 f which is Eq 81 Ob and the basis for the nondimensional friction loss curves valid for plastic pipes with D lt 125 mm Derivation of Equation for AHC The difference between the minimum pressure head and the pressure head at the closed end ofa lateral AHC is used to calculate the minimum head in the lateral Hn This is because the pressure head at the end ofthe lateral is easily calculated as HCZHI hf Ahe where Ahe is negative for downhill slopes But the minimum pressure head does not necessarily occur at the end of the lateral when the lateral runs downhill Thus in general Merkley amp Allen Page 246 Sprinkle amp Trickle Irrigation Lectures nHC AHC 417 o The above is from Eq 227 in the textbook 0 These concepts can also be interpreted graphically as in Fig 221 0 Following is a derivation of an equation for AHc based on Keller and Rodrigo 1979 1 The minimum pressure in the lateral occurs where the ground slope for a uniform slope equals the slope of the friction loss curve The dimensionless friction loss curve is defined as Eq 810b or Eq 223b th X 418 hf L pair 2 The slope of this friction loss curve is hfx X 175 paquot 275 j 419 X L d L 3 The uniform ground slope on the dimensionless graph is Ahe SL 1oos 420 hf J39FL J39F pair 7 100 4Then 139FS 275y13975 421 in which y is the value ofXL where the minimum pressure occurs 0 s y 1 S is the ground slope mm J is the friction loss gradient forthe ow rate in the pair oflaterals m100 m and F is the reduction coef cient for multiple outlet pipes usually about 036 Sprinkle amp Trickle Irrigation Lectures Page 247 Merkley amp Allen ALL how P y 7 10 5 smverury 11 75 1005 422 V 2 75J F on 11 75 y N 1098 423 J WhErEFz as MemevaAHen Pagem 6 Referring to the figure on the previous page the following equality can be written 275 AHC 100yS I 424 hf pair J F solving for AHC 10035 275 AHC hfpa JF y j 425 where J39FL fpak EE 426 and y can be approximated as in step 5 above for F 036 7 After manipulating the equation a bit the following expression is obtained 42m AHC 89LS13957 J39 3957 for AHC in m L in m Sin mm and J in m1OO m Note that J and L are for the m of laterals not only uphill or only downhill Ill Derivation of Equation for or o The parameter 0c is used in the calculation ofinlet pressure for a pair of laterals on sloping ground where Eq 2217 X HI 2 Ha pair E pair With q 1x H a 429 a Kd 43m Sprinkle amp Trickle Irrigation Lectures Page 249 Merkley amp Allen 1oos Ahepair F 431 hf pair Note that Ahepair must be a negative number The ratio XL is the distance to the manifold where L is the length ofthe w of laterals The following derivation is based on equations presented by Keller and Rodrigo 1979 1 Given that for a single lateral approximately of the friction loss occurs from the inlet to the point where the average pressure occurs multiple outlets uniform outlet spacing constant discharge from outlets single lateral pipe size we have the following 3 X 3 X althfPair Zlthfdownhill I Zlthfuphill1fj 432 The above equation is a weighted average because the uphill lateral is shorter than the downhill lateral 2 Recall that h X 275 i j 433 hf pair L Then X 275 hfdownhill I hfPair 2 75 434 h 1 X h fuphill TE fpair 3 Combining equations och h 5 5 275 1 1 5 275 435 fpair 4 fpair L L L L 3 X 375 X 375 or l 436 4 L L Merkley amp Allen Page 250 Sprinkle amp Trickle Irrigation Lectures y I 340 i 39 he on g maaww avg hW 1 pau U C Aim E g h39 E xL 1 xL 0 1 0 Tm as L Equater fur a 5 Eq 22 25 frumthe tEXIbEIEIk Seethe gure be uvv Pagezm h 0W 55 W Memey mm er 05AH 9 pair quotquot39v Ligh l f f f LU tI pair Merkley amp Allen Page 252 Sprinkle amp Trickle Irrigation Lectures
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