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"‘ .‘ “(M35 “ Maw-Ir '2‘?“ ‘- ' $3.153?“ #fi v 19' UV :. , v1 4» , n: *3 v. ,. n Y . - 7 -. . ,. ..._4, ~.~ ”mg-1.“. ‘ .1 « ‘-. ,. . 32m x ..,..,.- .. m“. ‘ a v. r: 'l- h.“ {.3 r '4' '~' m A. . . . ‘3’: I’a'nv ? .1 .. . 543-1 33% ,, . ”$52515? ’ , v r *7“: r: . _; up. " xfifi'fiai'la «- my -- 2‘ ..J..r;...‘...w w .- 1-. .n $2” ‘ 3-. «.s . J ‘V 18“.“ .7 . :‘w'fii‘s‘ ;. ‘L' '4‘"); “x, - 4-4- «g. '4 .. "'..-- " W ”37"“. ‘5 ‘ n 4 ‘ - r Date 21916151319 NIVE ERSITY LIIBRARlEsl 11111111111111111111111111111111111 1| 3 1293 00585 6012 LIBRARY Michigan State University This is to certify that the thesis entitled FINITE ELEMENT ANALYSIS OF DRIP IRRIGATION HYDRAULICS USING QUADRATIC ELEMENTS AND A VIRTUAL EMITTER SYSTEM presented by Shaun Francis Kelly has been accepted towards fulfillment of the requirements for Master___ degree in Mal Engineering we? MS U is an Affirmative Action/Equal Opportunity Institution PLACE N RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before due due. DATE DUE DATE DUE DATE DUE ’7', r . JAN: 0 f 119213 .1 ‘ .1 " l.’:,)\ )_ 1” . , r 1wa? a '3 blur MSU Is An Afflrmdive Action/Equal Opportunity institution FINITE ELEMENT ANALYSIS OF DRIP IRRIGATION HYDRAULICS USING QUADRATIC ELEMENTS AND A VIRTUAL EMITTER SYSTEM BY Shaun Francis Kelly A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE in Agricultural Engineering Department of Agricultural Engineering 1989 b040172. ABSTRACT FINITE ELEMENT ANALYSIS OF DRIP IRRIGATION HYDRAULICS USING QUADRATIC ELEMENTS AND A VIRTUAL EMITTER SYSTEM BY Shaun Francis Kelly As use of drip irrigation increases, the need to analyze large systems becomes more and more apparent. The purpose of this research was to conserve water and energy through the improved design of large drip irrigation systems by using the finite element method and a virtual emitter system. A differential equation describing flow in a drip irrigation system was developed and solved by using classical finite element methods. Several different methods were found to incorporate emitters in a virtual node system, where several emitters are combined into one element or node, as a point source. The virtual emitter methods were evaluated using linear and quadratic elements and found to be fast and efficient methods for analyzing the hydraulics of drip irrigation systems. ACKNOWLEDGEMENTS Sincere thanks to, my committee members, Dr. Larry Segerlind, Dr. Roger Wallace, and Austin Miller for their guidance, comments and involvement with the preparation of this thesis, to Dr. Vincent Bralts for the opportunity, the support, the encouragement, the enthusiasm, and the patience, and especially to my parents. ii Table of Contents LIST OF TABLES - v LIST OF FIGURES vi LIST OF SYMBOLS - - ix 1. INTRODUCTION 1 1.1 Scope and Objectives ................................................................................. 5 11. REVIEW OF LITERATURE AND THEORY 9 2.1 Hydraulic Design of Drip Irrigation Systems ............................................ 9 2.1.1 Hydraulics of Emitter Flow ................................................................ 9 2.1.2 Hydraulics of Lateral Line Flow ......................................................... 10 2.1.3 General Lateral Design ....................................................................... 17 2.1.4 The Statistical Uniformity Concept .................................................... 23 2.2 Network Analysis Techniques ................................................................... 26 2.2.1 Hardy-Cross Method .......................................................................... 27 2.2.2 Newton-Raphson Method ................................................................... 29 2.2.3 Linear Theory Method ........................................................................ 30 2.2.4 Finite Element Method ....................................................................... 31 2.3 Numerical Solution Techniques .................................................................. 40 2.4 Summary and Discussion ........................................................................... 42 III. METHODOLOGY 44 3.1 Research Approach .................................................................................... 44 3.2 Theoretical Development ........................................................................... 47 3.2.1 Differential Equation for Pipe Flow ................................................... 48 3.2.2 Differential equation for pipe flow including emitters ...................... 52 3.2.3 One-dimensional linear element using Galerkin’s method ................ 56 3.2.4 One—dimensional quadratic element using Galerlcin’s method ........... 57 3.2.5 Emitters as point sources .................................................................... 58 IV. RESULTS AND ANALYSIS - - - ............. 61 4.1 Computer Model (V NODE) ....................................................................... 61 4.2 Analysis ...................................................................................................... 63 4.2.1 Accuracy and Speed of Convergence ................................................. 66 4.2.2 Initial Estimates and Convergence ..................................................... 73 4.3 Summary Discussion ................................................................................... 75 V. CONCLUSIONS AND RECOMMENDATIONS 76 APPENDIX A Computer Model (VN ODE) -- - - 73 APPENDIX 3 Sample 0am 8‘? APPENDIX C data - - -- - - - - ‘3 LIST OF REFERENCES - - - - 98 iv List of Tables Table Page 1. Coefficients for equation [55] ......................................................................... 56 2. Accuracy and speed of convergence for the 12 solution methods .................. 67 List of Figures Figure Page 1. A typical drip irrigation system layout and components. ................................ 4 2. A submain unit with laterals and emitters ....................................................... 11 3. Moody Diagram ............................................................................................... 15 4. Comparison of Hazen-Williams vs. Darcy Weisbach. .......................... 16 5. Approximation of pipe friction loss. ............................................................... 18 6. Shape of the dimensionless energy grade line. ................................................ 20 7. Pressure variations due to various situations. .................................................. 22 8. Dimensionless energy grade line extended to the submain unit ...................... 24 9. Straight pipe element ....................................................................................... 33 10. Three successive pipe elements ...................................................................... 34 11. Two pipe elements and an emitter element. ................................................... 37 12. Example submain unit .................................................................................... 38 13. Example solution matrix ................................................................................. 39 14. Virtual node substructure ............................................................................... 46 15. Pipe element ................................................................................................... 49 16. Pipe element with emitters ............................................................................. 53 17. Pipe element with emitters as point sinks ....................................................... 59 18. Flow chart for VNODE .................................................................................. 62 19. Lateral line ...................................................................................................... 64 20. Pressures along the lateral line with varied node numbers. ............................ 65 21. Exact solution vs. seven node linear approx. (no point sources) .................... 69 22. Exact emitterpressures vs. seven node linear approx. pt. sinks ..................... 7O 23. Exact emitter pressures vs. seven node quadratic approx. ............................. 71 vi 24. Exact emitter pressures vs. seven node quadratic approx. (point .................. 72 25. Iteration vs. maximum defect. ........................................................................ 74 vii goo99>np p W} {H"’l {H} h, LIST OF SYMBOLS pipe constant pipe constant, equation [7] cross sectional area of pipe flow linearized emitter constant linearized pipe constant diameter of pipe Jacobian matrix, equation [21] length of pipe element coefficients for differential equation number of emitters per plant element force vector global force vector dimensionless friction factor reduction coefficient to account for emitters acceleration of gravity 6.2 coefficients for differential equation element nodal vector global solution vector pressure head average pressure head in the element headloss due to friction viii headloss due to friction in a lateral line pressure head for a given length ratio total friction loss total friction loss due to elevation friction loss in the lateral line pressure head at node i pressure head at node j pressure head at node k initial pressure estimate pressure head at the origin, equation [13] length ratio 1 element stiffness matrix global stiffness matrix emitter constant friction loss coefficient, equation [23] constant in Christianson equation [9] modified pipe constant length of pipe pipe flow exponent iteration number number of emitters number of emitters interpolation function for node i interpolation function for node j interpolation function for node k emitter discharge Q'Sl 4t mean emitter flow coefficients for differential equation flow correction flow rate emitter flow rate emitter flow in the lateral line emitter flow in the submain unit flow in the first emitter initial flow estimate total flow in the submain unit Reynolds number energy drop ratio head loss due to elevation lateral head loss due to elevation head loss due to friction lateral head loss due to friction submain head loss due to elevation submain head loss due to friction element residual vector global residual vector standard deviation of emitter flow statistical uniformity coefficient as a percentage average pipe flow velocity coefficient of variation of the emitter constant coefficient of variation of the hydraulic pressure coefficient of variation of emitter plugging coefficient of variation of emitter flow average pipe fluid velocity emitter discharge exponent emitter discharge exponent elevation head elevation solution vector, equation [20] xi I. INTRODUCTION For most of the world’s pOpulation, agricultural production has been the major source of food for the last 4000 years. Historically the practice of irrigation has been an important part of food production for much of the world, especially in arid regions. With the world’s population reaching an estimated 6.1 billion people by the year 2000, experts are predicting an increasing world food crisis. In recent years this crisis has become increasingly evident in some developing nations. In Africa, prolonged drought and severe lack of developed water supplies, coupled with high population density and the highest papulation growth rates in the world, have triggered a crippling famine. This is an area which even in the absence of drought cannot produce enough food for its own population. Irrigation systems will continue to play an important role in food production as we approach the year 2000, when the world will need fifty percent more food than today to secure at least the current state of nutrition (Holy, 1977). Basically agriculture can be expanded in two directions - extensively by expanding cropland, and intensively through improved plant species and hybrids, irrigation, improved management schemes and proper use of chemicals and fertilizers. Irrigation is important for both expanding the available cropland and increasing production on current land. Worldwide, irrigated area has increased from 95 million hectares in 1950 to approximately 250 million hectares in 1985. This represents about 20% of the world’s total cropland and accounts for over 40% of the total crop production (Power, 1986). It has been estimated that by the year 2000 the total area under irrigation will be 500 million hectares (Holy, 1977). Expansion of irrigation will not continue unchecked, however, as freshwater resources are in short supply. In the United States we have become increasingly aware of diminishing water resources. Development of irrigation between 1960 and 1978 in the south central plains of the United States resulted in an explosion in feed grain production. This had a huge impact on the local economics, as feedlots and meat processing plants moved in. In 1958 there were 300,000 cattle on feed in Texas; by 1978 there were more than 4.9 million. Increased pumping demands on the vast Ogallala aquifer underlying this region led to dropping water tables, threatening the economy on the high plains. In five states where levels are dropping most pervasively - Arizona, Kansas, New Mexico, Oklahoma and Texas - net irrigated area has actually declined between 1978 and 1982 by 678,000 ha. or 14% (Postel, 1985). Of all the world’s freshwater resources used today, 80% is used for irrigation, with an average efficiency of only 37% (Power, 1986). By increasing the water use efficiency of our irrigation systems, not only can we case our ever-increasing demand on these water resources, but we can reduce the runoff of both water and soil. This can be achieved by improving more traditional irrigation techniques such as surface flooding and furrow irrigation. Efficiency can also be increased through improved management with irrigation scheduling techniques, and through use of new technologies in sprinkler design and drip irrigation. Drip irrigation principles evolved in the early 19005 from experiments using drainage tiles for subsurface irrigation. The drip irrigation systems popular today were introduced for use in greenhouses in the early 1940’s in England. Drip irrigation did not become widespread in practice until the advent of relatively inexpensive plastic pipe and emitters which became available in the 1960s. These allowed the development of economically feasible systems. Growth of drip irrigation usage has continued, with Australia, United States, and Israel leading the way. In 1980 the total area under drip irrigation in the United States was over 200,000 hectares. Drip irrigation techniques are being used in cotton and other traditional row crops, as well as in the landscape industry. Ninety-five percent of all irrigation in Michigan is drip or sprinkler. Most is used on fruits, ‘ vegetables and orchard crops, accounting for up to 250 million dollars (Michigan Ag. Statistics, 1987). Despite this impressive expansion in the United States, drip irrigation represents less than 1% of the total irrigated area worldwide. Because of high capital costs it has primarily attracted growers of high-valued orchard crops, grapes and vegetables. These new technologies are to costly and complex to benefit many farmers in deve10ping countries, but incorporating the principles behind these techniques and technologies could prove to be worthwhile. Drip irrigation is the practice of applying water using low application rates and low pressures from emission points located near the plants’ root areas. Water flows from the emission points and through the soil by capillarity and gravity. A drip irrigation system offers many unique advantages for efficient use of water and labor over conventional systems. An efficient drip irrigation system applies water close to the consumptive rate of the plants, or stores water temporarily in the soil depending on the growing conditions desired. Labor costs can be reduced because of the ease of automation and the use of chemical and fertilizer injection. Drip irrigation offers the farmer greater control of water delivery, and frequent inigation maintains a stable soil moisture condition that enables the farmer to irrigate with water of a higher salinity. A conventional drip irrigation system, (Frgure 1), consists of a water supply and a pump, followed by a network of mainlines, submains, laterals, emitters, flow and pressure controls and filters. From the mainline, water is delivered to the various irrigation zones. Within each zone (20 - 50 ha) there are commonly a number of submain units (1 - 5 ha) with manifolds that feed water to the lateral. Emitters are placed along the lateral lines to dischargewater near the bases of the plants. Anna 6:88 £5.8qu0 93 593. 529? :3qu etc 32.5 < A 8.65. mm mm. \ Wm mw~ \ /. a 532-. am. n V7 .8 73 s“ \ aw? \ «cutie 011W \ 39: 5.2: \IIN 5:: 3W\ 4.53., 55: u>3<> “ _ \ ~35, 55: . \ 3.8 2.52 25.—M cup“: , gowzo ,\. . E I ~ , mes 80.0 000.0 0.0.0 , as... . v0.0 a? la '1'!“ ”99311.11 I. Figure 3. Moody Diagram (Benedict, 1980) 16 l: 0.06 LASIUS : 2 0.05 . \ etzosx a SOLOMON tiers)- Q ~ < 0.04 - ‘ IL {It f'lzo ~ 1!“ ' Jo ‘ E ' W‘\ 9 0.02 - \ C IL um Fungi TURBULENT —-. 0.0' 3 r 1 1 ‘ 3 lO 2 4 IO IO REYNOLDS NUMBER an N .0 O u I Figure 4. Comparison of Hazen-Williams vs. Darcy-Weisbach (Howell, et al.) 17 2.1.3 General Lateral Design For lateral design, water flows through the entire lateral and gradually decreases as water is discharged along the length of the lateral. It is important to be able to calculate the head at any emitter on a lateral. Many methods have been used to calculate these pressures. A close approximation of a pipe which loses water along its length (as in a drip lateral) in relation to the loss from an identical conduit which discharges water only at the end of the line is (Goldberg, et al. 197 6). h, [8] where ho, h( are defined in Figure 5 A more precise procedure was suggested by Christianson who computed FKLQ" [9] hf = D28. where _ 0.5 10 F: l +_l_+(m 1) l ] m+l 2n 6N2 and m = velocity exponent use 1.85 for H—W 2.00 for D-W N = number of emitters n = 1.1 K = constant 18 head loss 9.5 3.... lateral line E—-—> q r .L r r—t—t—t—t—t—t—r‘t—t—t—j Figure 5. Approximation of friction loss in a lateral line using identical conduit which discharges water only at the end of the line. 19 On this basis it is possible to calculate the friction loss as though all of the water were flowing through the entire length of the pipe, then multiply by F, a reduction coefficient to compensate for discharge along the length of the pipe. As the number of emitters increases, the value of F approaches 0.33 which yields the same result as [8]. Wu and Gitlin (1975) presented the dimensionless energy grade line to approximate head loss at any point along any lateral line. Bralts, et al. (1987) expanded the concept to analyze manifolds. The following development follows closely that presented by Bralts, et al.(1987). The dimensionless energy grade line procedure assumes that all emitters along a lateral line discharge the same flow. The shape of the energy grade line can be expressed dimensionlessly to be the energy drop ratio, R,, as shown by Wu and Gitlin (1975). R _AH.-_1 (1 .)...1 [“1 i-AH - l where i =the length ratiol/L(Figure 6) m = velocity exponent 1.85 for H-W 2.00 for D-W and LAH.,AH are defined in Figure 6 The shape of the dimensionless energy gradient line is shown in Figure 6 for various flow conditions. The total pressure along a lateral line can be expressed by summing the original pressure, the pressure head loss due to friction, and the pressure head loss or gain due to elevation. Wu, et al.(1979) developed the equation 20 0 ._ COMPLETE TURBULENCE. ROUGH PIPE 0.1.\ 0.2... - - .. -TURRULENT FLOW IN SMOOTH PIPE WEN-WI LLIAMS EQUATION e—e—e— 0.3.. 0.4.. 0.5.. 0.0L 0.7.. PRESSURE DROP RATIO. R; (Alli/AH! Lo l r r L L L 0 0.1 0.2 0.3 0.4 0.5 0.0 0.7 0.0 0.0 1 .0 LENGTH RATIOJIL Figure 6. Shape of the dimensionless energy grade line (Wu, et al. 1974). 21 h, =H,-R,AH:I:R,.’AH’ [12] where h. = pressure head for a given length ratio H, = pressure head at the origin RM = pressure head loss due to friction Ri’AH' = pressure head loss or gain due to elevation Figure 7 shows the pressure variations due to various situations and the resulting energy grade line along a lateral. Bralts, et al. (1987) combined the general emitter flow equation, [1], with [12] and divided it by the flow equations for the first emitter, q, (q, = KI-Io‘). The resulting equation relates emitter flow anywhere along a lateral independent of the coefficient, AH AH’ ' 1131 959114.11,“ Jim-1 H. B . where all the variables are as previously defined. K. The dimensionless energy gradient line concept can also be used for drip irrigation submain manifolds. To approximate head loss along the manifold, lateral lines are considered similar to uniformly spaced emitters. Using the same concepts as for lateral lines, Bralts, et al. (1987) presented the following, equation. Wilklflki )iR{A:]iR{A:Z]] where RAH, = manifold head loss due to friction A”: ”- RAH, = lateral head loss due to friction R,'AH,’ = manifold head loss or gain due to elevation RfAI-I,’ = lateral head loss or gain due to elevation "I”! LO” 0' PORT" ”HAL "I”! Datum!“ "I”! LOSS IV non "IS-JR! MAD Mt”! L08 IV "W ”not. "I”! 061000?" POEM W I! m J r L r r r 1 1 r I 0 0.! 0.3 0.3 0.0 0.0 0.0 I.) 0.0 R. 1.0 MW um A Figure 7. Pressure variations due to various situations (Wu, et al.). 23 q, = total flow into the submain unit it = flow equation exponent 4,, = flow anywhere in the submain unit Equation [14] can be used to approximate emitter flow at any point in the submain unit once the basic parameters of the submain unit hydraulic system are known. This is shown schematically in Figure 8. The dimensionless energy grade line concepts presented are the best approximations to flow rates in a drip irrigation submain. It should be realized that these are only approximations because the procedure assumes all emitters flow at the same rate as well as assuming the relationship between flow and pressure in a lateral behave the same as an emitter. Equation [14] must also be used with caution because the exponent x is the same for lateral line and emitters alike. To obtain better estimates of flow, network analysis techniques need to be used. 2.1.4 The Statistical Uniformity Concept The statistical uniformity concept consists of a statistical approach to the uniformity of emitter flow and irrigation application efficiency based on the coefficient of variation. The coefficient of variation gives a normalized measure of dispersion which can be used to compare the relative variation of different systems and is defined as the standard deviation divided by the mean of the quantity of interest. Bralts, et al., (1981) recommended the use of the statistical uniformity coefficient for drip irrigation lateral lines and can be defined by 0° LO ’9’ .9 ‘ Virtual Rouge 1 ¢ e. 9 7 . e V . s‘ I‘fi. ‘0 . I ~“‘----.—'/ c’ s /’ ‘~ ”~-— 1"," ~¢‘ ‘ ‘~~‘~~~-—‘ , 0 ~. 02 ‘- ~‘~ -—'d’ ~ ~-- ' '. ~“ ~-’-—"' ~‘~ ~Inn-..—‘“"' ‘ ‘~ I I ~-" 3 Dimenetenet Preceure slut-cc Lateral Line Length letie Figure 8. Dimensionless energy grade line extended to the submain unit. 25 1 U, =100(1-Vq)=1oo(1_§q_1] 1 5] where U, = statistical uniformity coefficient as a percentage V,l = the coefficient of variation of emitter flow Sq = the standard deviation of emitter flow 3 = the mean emitter flow rate The variation of emitter flow in a drip irrigation submain unit is the result of a variety of factors. The primary factor is hydraulic design. Other important factors are emitter type, emitter manufacturing, emitter plugging and the number of emitters per plant. Bralts, et al., (1987) shows how the variation of emitter flow in the submain unit can be determined as a probabilistic combination of these relevant factors. .1 2 2 2 2% [16] Vq=e ’(V,+v,+x V,) where V9 = the coefficient of variation of emitter flow V, = the coefficient of variation of emitter plugging V.I = the coefficient of variation of the emitter constant V.I = the coefficient of variation of the hydraulic pressure e = the number of emitters per plant It is important that emitter flow variation or the uniformity of water distribution be known when a drip irrigation system is designed. These quantities can be calculated using the above equations to evaluate the accepability of any particular system. For drip irrigation systems the general criteria for an acceptable statistical uniformity, U,, are 90% or greater, excellent; 80—90%, very good; 70-80%, fair; 60-70%, poor, and less than 60%, unacceptable. 26 2.2 Network Analysis Techniques As stated earlier, the flow in a drip irrigation system can be considered to be hydraulically steady, spatially varied, one dimensional, incompressible pipe flow. Methods for approximating the flows and pressures were presented in the previous section. Network analysis techniques need to be used to more accurately describe the spatial variability of flow and pressure in such systems. The solution of any network problem must satisfy both continuity and energy principles. The continuity principle states that the net flow rate in any piping junction must be zero. Energy principles state that the not head loss around any loop of the network must be zero. Because of the non-linear relationship between flow and head it is not easy to calculate the flow and head at each point in the system. Network analysis techniques are generally implemented on a computer because of their iterative nature and the need for speed and accuracy. The most popular computer—based network analysis techniques used today are based on Hardy-Cross, Newton-Raphson, and linear theory techniques. The use of these techniques by irrigation engineers for the design of drip irrigation systems has been limited. Solomon & Keller (1974), Wu & Fangmeir (1974) and Perold (1977) have used iterative techniques based On the backstop method. Bralts & Segerlind (1985) proposed a method based on a linearized set of flow equations using the finite element method. All of the network analysis techniques described here can be derived from a linearization of the system of energy equations. The non-linear terms in the energy equation can be written in terms of flows or heads and expanded into a Taylor’s series in the following manner (Wiggert, 1985). The function F(Q) contains the non-linear terms of the energy equation. 27 _ 2 [17] HQ) =F(Q..)+F'(Qa) (Q -Qa)+F”(Qo)£2Q—O)+ or in terms of the pressure heads F(h). (fl-H”)2 [18] -——+... F (H) = F (11,) + F '(H,) (H — H.) + F ”(H,) 2 where H, and Q0 are estimates of the pressure heads and flows at the point the solution is expanded about. It is usual to consider only the first two terms of the series expansions in the above equations to obtain a first order approximation of the non-linear terms. The Hardy-Cross, Newton-Raphson and linear theory methods will be presented in the following sections. 2.2.1 Hardy-Cross Method The Hardy-Cross method was one of the first methods developed to analyze hydraulic pipe networks. This method was popular in the pre-computer days as a hand worked solution, but is still used in various forms as the basis of many computer algorithms for the solution of hydraulic network problems (Edwards and Spencer, 1972). This loop method, first used by Hardy-Cross, is an iterative flow corrective technique using assumed pipe flow rates, based on continuity to solve the energy based loop equations of a hydraulic network. Flow adjustments are made according to the following equation which can be derived from the Taylor’s series expansion [17]. [19] 2}! Q Q. Q "2% where XI: = net head loss around any closed 100p (m) n = the flow equation exponent 1.852 for H-W 2 for D-W Q = the flow through an individual pipe (m3/s) Barlow and Markland (1969) used a second order approximation of the above equation for corrective flows and obtained a more rapid convergence. The Hardy-Cross iterative solution is outlined in the following steps following Wiggert (1986). 1. Assume an initial estimate of the flow distribution in the network which satisfies continuity. An initial estimate closer to the correct solution results in faster convergence. 2. For each loop evaluate Q with [19]. The numerator should approach zero as the loops become balanced. , 3. Update the flows in each pipe in all the loops using the following equation. Q0: +1) = 9(5) + AQo.) [20] where n = the iteration number Q = the corrected flow AQ = the flow correction from [19] 4. Repeat steps 2 and 3 until a desired accuracy is attained. The Hardy-Cross method is a simplified version of the method of successive approximations. A modified version of this method called the nodal method was also used by Barlow & Markland (1969). In this method the heads are solved for instead of the flows. The heads are successively approximated at each node using an equation similar to [19]. It can be derived from [18] and is given as 29 [21] i {age where Q is the sum of the flows into or out of a node and n, h and Q are as defined earlier in [19]. The solution follows the more traditional Hardy-Cross method only the heads are corrected instead of the flows. The Hardy-Cross method can easily be used to solve relatively small networks using either a hand held calculator or spreadsheet algorithm on a microcomputer. The continuity relations are in a sense "decoupled" from the solution of energy equations, they are satisfied initially with the assumed flows and remain satisfied throughout the solution (W iggert, 1985). The fundamental drawback of the Hardy-Cross method is poor convergence due to the independent solution of loop and nodal equations. 2.2.2 Newton-Raphson Method The Newton-Raphson method is a method of successive approximations similar to the Hardy—Cross method (Jeppson, 1982). The Hardy-Cross method solves each loop equation one at a time during each iteration. The Newton-Raphson method solves all of the loop equations simultaneously determining all of the corrective flows and head values in the network each iteration. Convergence in this case is quadratic. The Newton-Raphson method commonly used in hydraulic network analysis solves for corrective flows using the following equation derived from the Taylor’s series expansion of corrective flow equations. AQQ +1) = Agar) - Z. [22] where AQ_,1 = the vector of newly estimated corrective flows AQ, = the vector of previously estimated corrective flows 30 Z. = the solution vector of the linear system of equations 0.2,, = FIll where F l = matrix of corrective loop flow equations and D = the Jacobian matrix i 31?, 317, at", ' [23] BAQI BAQ: BAQ. arr, arr, ar, Bag, 3139, am, 3F, 8F, 8F, .BAQn BAQz 3AQ.J The Newton Raphson technique can also be used to solve a set of nodal equations for corrective pressures, (AH). In this case AH is substituted in place of AQ in the above equations. The convergence of the Newton-Raphson method is highly dependent upon a reasonable first approximation (Jeppson, 1977). Another disadvantage of the Newton-Raphson method is the need to evaluate first derivative terms in the Jacobian matrix D each iteration. 2.2.3 Linear Theory Method The linear theory method of network analysis was first proposed by Wood & Charles (1972). This method has several advantages. The most significant is that convergence to the final result is rapid and does not require initial flow estimates or complicated differential equations for the solution. Network analysis theory method solves both loop or nodal equations. The basic theory transforms the loop or nodal equations into a set of linear equations that can be 31 easily be solved. An initial approximation of the flow rates is generally obtained by assuming laminar flow exists. The calculated flow rates are used to determine the coefficients in the equations each successive iteration. The transformation of the loop energy equation results in 2"! = “up”? = mo), .{ = 0 [24] where Qll = approx. discharge K‘ = modified pipe constant The nodal equations can also be transformed in a similar manner that results in a highly accurate solution obtained with few iterations. However, in some cases convergence was never obtained. By far the most common application of the linear theory is with the loop equations. 2.2.4 Finite Element Method The finite element method is a systematic numerical procedure used to solve many complex engineering problems. This method has been used with discrete elements such as in structural analysis and continuum element problems such as groundwater movement and heat transfer. The classical finite element method involves as integral formulation and a set of piece-wise smooth equations to approximate a quantity. The use of the finite element method for the solution of hydraulic network problems is a simple extension of the original development for structm'al assemblages (Bralts, 1987). When laminar flow is present throughout the hydraulic network, the relationship between head loss and flow is linear and was analyzed using the nodal equations and the finite element method. This special case of the linear theory method was solved using the finite element procedure by Norrie & DeVries (1978). Bralts & Segerlind 32 (1985) used the finite element method for network analysis of a drip irrigation system. The following is presented from their development of what a single pipe element contributes to the continuity equation. Friction losses in tees and elbows were neglected. The energy equation is written for a straight pipe element, (Figure 9), and rearranged into the following equation. Q = C,(H, —H,.) + C,(z,. -z,.) [25] [I (Z.- + H.) - (Z,- + H;)l (If-2] where p = ké H, & 1‘1j = upstream and downstream static pressure heads Q = flow through the pipe k = friction loss coefficient m = exponent in D-W or H-W The finite element method utilizes the concept of an element stiffness matrix and element force vector to construct a system of equations. For element (e),(Figure 10), the system of nodal equations for s and t are —Q.‘""+Q.‘"=0 ”61 -Q.‘"+Q."”’=0 [2"] where flow into a node is negative and flow away from a node is positive. The contribution of element (e) to the above equations is Qf"=C,(H.-H,)+C,(z,-z,) [28] and .‘"=-C,(H.-H.)-C,(z.-z.) [291 The element matrices are the contribution of an element to the nodal equations that it touches, 33 21 Figure 9. Straight Pipe Element xi— \/ Figure 10. Three successive pipe elements 35 Q5" [0, -C,HH:} {0,132} [30] Q1“) — -CP Cr H - {9A2 where AZ =2, -z,. Equation [30] has the standard finite element form {W} = IX“) {11“} - 0‘") [311 where [Kan _[ Cr -Cr] ' [32] - -Cr Cr is the element stiffness matrix. H [33] (c = ' {H i {H} is the vector containing the element nodal values and C AZ [34] {#ng AZHg} , '8 is the element force vector. The product of C, Z is g and all other variables are . previously defined. The element matrices are assembled using a direct stiffness algorithm (Segerlind, 1984) and yield a system of equations which have a general matrix form as [Kl {H} - {F} = {0} [35] where [K] is the global stiffness matrix {F} is the global force vector {H l is the vector containing the nodal pressure values for the network being solved. 36 Emitters in a drip irrigation system were implemented in the system by considering them to be a separate element (Figure 11). Q = C¢(Hr‘ -Hj) [36] where C, is the linearized coefficient for the generalized emitter flow equation as presented by Wu et al. (1979). The flow equation for an emitter junction (Figure 11) is Q:"1)+Q,("”=O [37] An emitter head connected to node sis incorporated into the final system of equations as follows. First add the value of C. directly to the diagonal value K“ in [K]. Second add the value of CJ-l, directly to rows of {F }. An example submain unit layout and the associated stiffness matrix was presented by Bralts (1987) and is shown in Figures 12 and 13. Bralts and Segerlind (1985) developed a computer program to solve drip irrigation systems using the finite element method described above. They cited numerous advantages. The final system of equations is symmetric and banded. Proper numbering of the pipe junctions and emitters resulted in a relatively small bandwidth. A large drip irrigation design problem can be stored and solved in a relatively small computer (Bralts, et al. 1987). Another advantage of the finite element method is that once the equations are developed in the standard finite element form [31], then many of the existing finite element programs can be used to solve and assemble the equations. 37 (e) Figure 11. Two pipe elements and an emitter element 38 ' O 50 9? U) (5) (9) 2.1 65 10 q, (2) (6) (IO) 3., 7$ n (a (3) (7) (ll) (4) (8) ('2) t O 4 a 12 19 Figure 12. Example submain unit STIFFNESS MATRIX b m m cue. 4:. l "1 m m.m (at 4:. c‘ c' 4:. H2 oe. m m m m -Cl 9“. CI H3 06! m m mi m c. ‘0.“ H‘ (5| l6! f9.“ 0 H5 1_ m 3.3. m Go Go “5 B&_— “l I! Hi (7| ‘0 to G. ‘. H7 L.& M II "I Hi 4:. a. to? 4:. H8 4§L II (I @009 40 H9 ‘ 3'3 H W 10 £0 ,9 .—_ 1 IN" 9°C. to: MI "2.1 H12 4:. 4:0 «:0 L ‘ .J Figure 13. Example solution matrix (Bralts and Segerlind. 1985) 39 vecron or menu FORCE L W r ,m r ,tzl.,m .01-,(2) ,m-,m .15)-,14) . .16)-,(51 ,m-.te) ,ra)-,rn .19)-,m ,no)-,m ,nu-.tro) ,nzr-,nu . 40 2.3 Numerical Solution Techniques Use of substructures and various techniques for solving the resulting set of equations will be described in the following sections. 2.3.1 Virtual Nodes in the Finite Element Method Bralts and Segerlind (1987) proposed a technique for incorporating a virtual node concept using the finite element approach to drip irrigation system hydraulic design. They proposed that this technique could allow the concentration of the solution matrix into a simple, more compact virtual node matrix. This solution technique could possibly improve the speed of convergence with large drip irrigation systems of 10,000 or more emitters. The motivation for this section is to find the scope of current use'of such techniques and determine their applicability of such techniques to solving large drip irrigation network problems. To analyze a large complex structure it is necessary, for economic, computer capacity or organizational reasons to divide it into a number of smaller substructures. Substructure coupling procedures have been developed for finite element analysis of large structures. Each substructure is analyzed as though it were a separate problem. The various substructures are then coupled together in such a way that equilibrium and continuity are satisfied at common boundaries. Substructures have been used to analyze the separate components on an airplane where natural substructures occur, such as fuselage, wings and tail. The overall problem then involves only the variables on the boundaries of the substructure. The idea of the substructure may be nested. Each substructure may itself be substructured to any depth and identical substructures could greatly reduce calculations. Artificial substructures can also be introduced. A two dimensional problem consisting of a regular q x q grid of 41 square elements could be solved by dividing the elements into four groups with a " + " shaped cut, then dividing each group into four parts again with a " + " shaped cut and so on. This was termed "nested dissection". Utilizing a technique similar to substructuring, a large drip irrigation system could be analyzed using the finite element method. It could increase the speed of convergence as well as solve some of the problems that occur when storing such a large network with limited memory. 2.2.2 Solution of Linear F‘mite Element Equations Once the energy and continuity equations for network analysis are put in standard finite element form, [K] {H} - {F} = {0} [38] there are numerous methods for solving these sets of linear equations. Bralts and Segerlind (1985) solved these resulting equations for a drip irrigation submain unit. They based their initial estimates of the nodal heads using the energy grade line concept (Wu, et al., 1975). Using these estimates they calculated the linearized flow coefficient, C,, in [K] and {F}. [K] and {F} were used to solve for new head values, {H}, using Gaussian elimination techniques. These new head values were used to calculate updated C, values in [K] and {F}. The process was repeated until changes on successive head values, {H l , were small and the solution was considered to have converged on the correct solution. The principle direct methods for solving finite element sets of linear equations are all variants of Gaussian elimination. Two iterative methods that have come to prominence recently occur more frequently in finite element methods are the frontal technique and band matrix techniques. 42 It is thought that the use of these newer solution techniques could be useful when solving a large drip irrigation network on a small computer using the finite element method described by Segerlind and Bralts (1985). Some of these newer solution techniques could be applied along with a substructuring could be investigated using a large drip irrigation network. 2.4 Summary and Discussion The review of literature and theory was written to provide the theoretical framework and motivation for the proposed research. It is hoped that this provides a basis with which to improve the techniques available to analyze the hydraulics of large drip irrigation systems using computer based network analysis techniques. The review of hydraulics of drip irrigation systems was included, as it is the basis of hydraulic design of drip irrigation systems. The resulting non-linear pressure-flow relationships of emitter and pipe flow were presented as they occur in a drip system. Empirical relationships are generally used to calculate emitter flow and friction loss in pipes. Hazen-Williams and Darcy-Weisbach were used predominantly in drip irrigation design, with the Darey-Weisbach considered as more accurate. Various methods to get around the problems that the non-linearities of the pressure-flow relationship and the spatial variability in a drip lateral line were presented. Graphical and simplified computational approximations of head loss in pipes have prevailed for drip irrigation lateral design. The approximation technique that seemed best suited for computer use was the dimensionless energy grade line concept. This concept was expanded to include a whole submain unit and provides a good initial 43 estimate for head loss for use in other network analysis techniques. Network analysis techniques were presented and their use in drip irrigation design was found to be limited. All the methods use a linearized form of the energy and continuity equations that result in loop or nodal equations that are solved iteratively for flows or heads. The use of the finite element formulation of the nodal equations extends the methods of solution to the expansive techniques that are available in many finite element programs. The use of substructures for the solution of large drip irrigation networks was mentioned. The techniques could be applied to the finite element formulation of Bralts and Segerlind (1985). More efficient techniques are available for solving systems of linear equations such as the frontal method and sparse matrix techniques could be used to reduce storage requirements and speed convergence of large networks. Even with the speed and power of microcomputers seeming to increase every day, the utilization of efficient techniques opens the door for ever increasing analysis without the need for mainframe storage and speed. III. METHODOLOGY The review of literature has shown that network analysis techniques have seen limited use in the design off drip irrigation submain units. As use of drip irrigation grows, the need to analyze large irrigation networks with 10,000 or more emitters efficiently becomes more apparent. For row crops such as tomatoes, it is easy to imagine a submain unit of 1.5 ha with 100 m laterals spaced 1.5 m apart, with an emitter every 1 m. This system would contain 15,000 emitters. The goal of this research is to utilize the finite element method and network analysis techniques to reduce the number of nodal equations and obtain a virtual node system. Development of a procedure would lead to improved hydraulic design of drip irrigation systems to conserve water and energy. 3.1 Research Approach Based upon the need for comprehensive hydraulic design techniques for drip irrigation networks, the following approaches are proposed for achieving the major research objectives. Objective 1. Development of an equation to describe the flow phenomena in a drip ' irrigation system. 45 The approach to be followed under objective 1 will be to use conservation of energy and mass principles along with empirical equations describing the pipe friction losss and emitter flow to develop a general differential equation expressing the pressure head in a lateral line. Objective 2. Investigate the use of virtual node concept in hydraulic design of drip irrigation systems. The approach to be followed under objective 2 will be to develop the nodal equations for network analysis using a virtual node system. A virtual node system will enable large drip irrigation systems to be discretized into a manageable number of equations by combining emitters into virtual nodes Figure 14. After the nodal equations are developed, the finite element method will be used to solve for nodal pressures at each emitter. Objective 3. Define limits of application, efficiency of solution, improve speed of convergence, and improve initial estimates of the solution method using a virural node concept. The approach to be followed under this objective is the development of a finite element based computer model using the concepts formulated in objectives one and two. The computer model will be designed such that various emitter types, lateral line sizes, and submain units can be analyzed. Using the computer model, speed of convergence, efficiency of solution, accuracy, and initial estimates of the different methods developed 2:, ., 20.7 a , , z, 3,, [3% .’., ,’ ’9; ,I f , .. .4" a ,g . bmflzmig. i... an H . H STIP 3 371' I , gm , i . 2,, ,'0' 493 an 37" l Figure 14. Virtual node substructure. 47 will be compared to an exact solution of a drip irrigation system. Once the advantages, disadvantages and limits of application are determined, its use in hydraulic design of drip irrigation systems will be evaluated. 3.2 Theoretical Development The objective of the following development is to obtain a system of equations describing the flow phenomena in the submain, laterals and emitters of a trickle irrigation system. The resulting equations will allow the submain unit to be discretized into elements which can contain any number of laterals or emitters. By using conservation of energy and mass principles along with empirical equations describing pipe friction loss and emitter flow a differential equation can be developed to solve for streamline or potential flow in a submain unit. In streamline flow the quantity of interest is the flow rates or fluid velocities. For potential flow the quantity of interest is the pressure heads. In either case, once one of the quantities is solved for the other can be easily found using the empirical relations relating flow rates to pressure heads. The following sections derive a differential equation in terms of potentials or pressure heads in a drip irrigation submain unit. The differential equation takes the following form 8’]: [39] 0‘55 + Gh + Q = 0 where h, is the pressure head or potential anywhere in the submain unit. D,, G, and Q are coefficients in the differential equation and are functions of the pressure head and 48 pipe section. Equation [39] is solved using the classical finite element method by applying Galerkin’s formulation. The coefficients, 0,, G and Q are recalculated using the newest approximations for pressure head and an iterative solution is obtained. First, it is shown how a grid of one dimensional linear elements is used to approximate pressure heads in the submain unit. Emitters are incorporated into [39] in three different ways with the G and Q terms. The method is then extended to use a grid of quadratic elements. Finally, it is shown how emitters can be incorporated into the solution of [39] by treating them as point sources. 3.2.1 Differential Equation for Pipe Flow For the pipe element in Figure 15, the energy at point x, can be expressed using the energy equation for incompressible pipe flow as 2 [40] v1 E,-h+z+§é- Similiarly, the energy at point x+dx is 8h 32 1 3v, 2 [41] 3,..-(h-$..).( wring...) Using conservation of energy principles,the energy at point x is equal to the energy at point x+dx. The energy equation is written for the pipe element of length, dx in Figure 15 as v2 81: a: 1 [42] H..- (kg...) (231...) 2( we, where h, is the head loss due to friction his the pressure head (m) 49 x— h energvgrodeline h—ifid" \ i ‘\ \ . Jul-d" ‘ \Va ax 1‘ \‘ v; r \\ \ z’gfia \ \l 1 *de X - clearer“ 1’12“” 15. P199 50 z is the elevation head (m) v, is the average fluid velocity (m/s) (II is the length of the pipe element (m) Subtracting h, z, and vf/Zg from both sides of [42] gives 3h 32 1 3v, 1 av, [43] Ed! +5317: +E[vx$d1)+g[gdx]z -h, The headloss due to friction in the straight pipe element in Figure 15 is empirically calculated from the generalized equation, [7], discussed in the literature review and rearranged in terms of fluid velocities instead of flow rates as follows h, = avfdx [44] where h, = head loss due to friction v, = average fluid velocity dx = length of the pipe element and if the Hazen-Williams formulation is used. , 5.88 [45] a = Crasonssss where C = Hazen-Williams coefficient A =cross sectional area through which flow occurs m = 1.852 For the straight pipe element in Figure 15, the fluid velocity at point x and x+dx are equal, (v, = mm), and the first two terms of the energy equation, [43], are exactly the friction loss described by [44]. Equating them results in the following ,. 3h 32 [46] av, dx -&-dx +754! 51 Equation [46] is then solved for the flow velocity, v,, giving [1H *‘ 5 ax i ax ’ a a [47] The quantity 82/3x in [47] is the slope of the pipe and is generally known as the change in elevation per unit length. The quantity of interest in [47] is the headloss per unit length, alt/3x. Equation [47] is linearized in terms of air/3.x when an initial estimate of the headloss per unit length or Alt/Ax is provided. This results in a linearized form of equation [47]. , - n91]? 314?]: ‘- 5 Ar 81: i at a a The law of conservation of mass for steady state flow requires that the rate of fluid [48] out of the pipe element be equal to the rate of flow into the pipe element. If flows into the pipe are considered positive and flows out of the pipe considered negative the mass balance for the pipe element of Figtue 15 is av [49] VA (v, + 3 dz} where A is the cross sectional area through which flow occurs. Equation [49] algebraically reduces to Aav,_0 [501 at _ The flow velocity given by [48] is substituted into [50] and the differentials are evaluated resulting in [51] 0,3340, =0 52 where L: 2 al.—[Air [5] t-iAx a and 12'— 53 8 in 3.2.2 Differential equation for pipe flow including emitters Using conservation of energy principles, the energy equation is written for the pipe element of length, dx which is losing fluid along its length at a rate q. in Figure 16 the same as equation [42]. The generalized friction loss equation, [44], is related to the energy equation in exactly the same way as if no emitters were present. This results in the same equation for flow velocity, '2 1 [541 ,, _ LEE] " ."—".+l.[—31]' ’ ' 5 Ax 3x 5 ax a a If flows into the pipe are considered positive and flows out of the pipe considered negative the mass balance for the pipe element of Figure 16 is 3v, 4. [551 A 75-; — 0 where A =cross sectional area through which flow occurs q, =flow out of the element due to emitter flow The flow velocity given by [54] is substituted into [55] and the differentials are evaluated resulting in the following differential equation to describe the pressure head in the pipe element including emitters. 53 one rgy grade 1an pipe acme!“ with ern'm‘fi' 8’): [56] 4. D‘ax2+Q’-dx_o The outflow from the pipe element, q,, is related to pressure head in the element by the following equation for emitter flow. 4, = nkh" ' [57] where h = the pressure head at the emitter x. = emitter exponent k = emitter constant n = the number of emitters/element The emitter outflow can be incorporated into the differential equation, [56], using one of the following three methods. The average pressure head in the element, h’ can be calculated from previous estimates of pressure heads at both ends of the pipe element and emitters can be considered to flow at this average pressure head resulting in Q. _ - AX hereafter referred to as method 1. The emitter flow equation can be linearized in terms of pressure head, h, resulting in ’0 -l [59] Go.) =[1ka1_]h Ax hereafter referred to as method 2. The emitter flow equation, [57], can be expanded using the first two terms of a Taylor’s series and estimated as x.-l x [60] 00mg, =[fli’igL_]h +(1-x,)£k;(h’) ' 55 hereafter referred to as method 3. The resulting one-dimensional differential equation describing pressure heads in a pipe element can be summarized as follows. 61 D,%32+G(h)+Q,+Q,=0 [ l The coefficients 0,, C. Q,. and Q, for three different methods is summarized in Table 1. The solution procedure is summarized in the following steps. 1. The submain unit or lateral is discretized into elements by specifying nodal points as well as specifying coordinate values. Specify the approximation equation. In this case the one dimensional linear. element is used. Initial estimates for pressure heads, h, at nodes must be specified to calculate D,, G, Q,, and Q, for each element. Subsequently, D,, G, Q,. and Q, are recalculated using the most recent values of pressure heads. Using Galerkin’s method an equation is written for each unknown nodal value using D,, G, Q,, and Q,. The system of equations is modified to include values of constant pressme head. Solve the system of equations. Compare previous pressure heads with new values of pressure head. If the solution hasn’t converged go to step 3. . When the solution has converged calculate other quantifies of interest. ( i.e. flow rates in average emitter flows, nodal flows.) 56 Table l. Coefficients for [61]. Emitter Outflow D; G Q: Q: Method A M “7" _nk(h')" A AZ '-.‘:= 1 :[75] 0 Ar :13] a ma A[Ah]L:'= [____""(”')x'-I]h A A2 1.7.. 2 ‘3. a A. o 7 2.: a ma £[éfl]¥ [in—Wit}; (Ida-$0)" A [g]? 3 % AX Ax % Ax a ma 3.2.3 One-dimensional linear element using Galerkin’s method Equation [61] can be solved using the classical finite element method by applying Galerkin’s formulation and one-dimensional linear elements. The residual equations for each element are as follows (Segerlind, 1984) {R“’} = [m {M - W} [621 57 where h,- [631 =3 (. _l_9,[1 —] 9i? 3 [64] [kn-L -11+61 and +++=—-[33 ‘] 2 l The element matrices are then combined using the direct stiffness procedure and the final system of equations, [K] {H} - {F} = {0} [65] results. 3.2.4 One-dimensional quadratic element using Galerkin’s method Equation [61] can be solved using the classical finite element method by applying Galerkin’s formulation and one-dimensional quadratic elements. The residual equations for an element are as follows. {13“} = a“! {h“’} - {rm} [671 where [68] and 58 D 14 —16 2 4 2 -1 [69] [k“]=a‘ —16 32 —16 +0.1. 2 16 2 2 -16 14 -1 2 4 and 1 1 [70] ++—-.- 3.2.5 Emitters as point sources In this section it is shown how an emitter can be treated as a point source in a section of pipe. Consider the linear pipe element in Figure 17 with an emitter located at X,. When emitters are considered this way emitter flow, q, is no longer constant throughout the element as in the previous solution technique, but is a function of X. As before, the flow in the pipe without emitters is 3’}: [71] D‘s-x7 + Q, = 0 The solution to [71] is solved using the finite element method and a residual equation is formed for each element. The residual equations are then modified as follows to incorporate emitters as point sources. {R“’1 =1k“1{h“’1 - W} [72] For the linear element the modifications are as follows. h, . [73] =3 S9 13‘ energy g'ode line 31—33-de .\ '\ ‘\ \ Vu" 331): dX Va 1 . q ‘ ‘ \ dd 2.1de \ ax EA/ 2 red“ X xa x1 x lcmcfit with emlttcrs “5 point S e 1. D, 1 _ N3 MN, [k ]=— +orG, 2 L -1 1 ~,-.- ~,. 3“ =%'E[i]+aalfii_, For the quadratic element the modifications are as follows. [m =& GI. -16 32 —16 +110 NjN, N} MN, [14 -16 2] N? MN,- MN. 2 -16 14 ”WE NkNj N: x.“ [74] [75] [76] [77] [78] N,, NJ and N, are the interpolation functions for the element evaluated at the point X,. If the emitter is located directly on a node that is shared by one or more elements the contribution of Q, is divided according to the factor a. 61 IV. RESULTS AND ANALYSIS In the results and discussion section, the methods derived in the theoretical development are analyzed using the computer model developed as research objectives. Utilizing the results of this analysis, the virtual node concept is then evaluated for use in hydraulic design of drip irrigation systems. 4.1 Computer Model (VNODE) A computer program, (VNODE) was developed to incorporate the virtual node concepts with the finite element method to analyze the hydraulics of a drip irrigation lateral. (Appendix A). The computer program (VN ODE) solves the general hydraulic network problem posed in a drip irrigation lateral, where flow is considered hydraulically steady, spatially varied, one-dimensional, incompressible pipe flow. VN ODE was developed in Turbo Pascal Ver. 4.0 on an AT class microcomputer. The problem is first formulated as a differential equation to describe flow in the drip irrigation lateral as described in equation [53]. The solution of this equation then follows the classical finite element method where the coefficients 0,, G and Q are all functions of the pressure head, h. The solution steps are described in the theoretical development. Figure 18 shows a general flow chart of the program. The program can use one dimensional linear or quadratic elements and depending on the formulation used, the functions D,(h), G(h) and Q(h) can be changed. Friction loss is approximated using the Hazen-Williams formulation. The system of residual START hput nodd md olementd date end lnlfld promos A 62 Add to no using the drect stiffness procedure. A for /\ Yes N, N], Mt edeuoto or promos. Figure 18. Flow chart for VNODE. 63 equations are solved for pressure heads at each node using Gaussian elimination. Pressure head in the lateral can then be calculated at specific points of interest ( e. g. at emitters) using the interpolation functions for the element chosen. 4.2 Analysis It was necessary to define a drip irrigation system to analyze with VNODE to facilitate comparisons between solution methods. A lateral line was chosen as opposed to a submain unit to keep the comparisons simple. The drip lateral in Figure 19 was selected for initial analysis. This is a lateral that might be commonly found in orchards . in Michigan. The lateral line is polyethylene tubing (C=150) of 15.24 mm id. [0.6 in.] and total length of 250 m. [820 ft.]. Emitters are vortex type, It=9.14x10'7 [0.008], x,=0.5 spaced every 5 m. [16 ft.]. Emitter flow at design pressures is about 17 lph [4.5 gph]. To determine how many elements would be needed to approximate the "exact" pressures in the lateral line, the system in Figure 19 was subdivided using an increasing number of nodes. It was observed from Figure 20 that 7 nodes would be sufficient to approximate the pressure gradient along the lateral. From this observation it was decided that further comparisons would be made using this 7 node configuration. Lotord Doto Emitter Data Msammlam 1'?) K- 9.14x10 [0.008) danetor - 15.24 m |.d. {0.6 In. ) Morsm meat. 17 l/h hput promo, 30 m K- 5 he: } S7 V V V . . . . 37 v :2 v I/ \l l\ 250 m /1 Figure 19. Lateral line. 65 oovoz 3 ecu n .N 6.53.. o octatabmnnm [in] peoq “mud Figure 20. Pressures along the lateral line with various nodal subdivisions. 4.2.1 Accuracy and Speed of Convergence The accuracy and speed of convergence were analyzed for the system in Figure 19 for each method developed in the theoretical development. The emitter flow equation was incorporated using the three methods as summarized in Table 1. In addition emitters were added to the nodes as point sources or distributed flow for both linear and quadratic elements. The results of this analysis is presented in Table 2. All analysis were performed using the same computer with calculation time for actual data computation, not including the time needed to input or output the data to and from the program. In general it was found that by including emitters as separate point sources increased computation time significantly. This is due to evaluating the interpolation functions N,, N, and N, (for the quadratic element) at every point on the lateral line each iteration. By incorporating emitters a distributed flow along the lateral, the calculation time was reduced significantly without sacrificing accuracy. Accuracy of each method was evaluated by comparing the total flows in the lateral of Figtue 19 for each method. Since the most important design parameter in most drip irrigation systems is the uniformity of emitter flow it was calculated for each method. Table 2 shows the total submain flow calculated by each method and the exact solution. In general most methods overestimated the flow slightly. Coefficient of uniformity for emitter flow rate didn’t change much regardless of the solution method. The calculated emitter pressures were plotted against the exact solutions for each method in Figures 21-24. For most methods there was no major deviation in estimated pressures. With the linear method when emitters were added as point sources for method 2 and 3, major deviations occured. It is not sure exactly why these occured only in these cases. These errors could have been caused by the G (h) term evaluated for point somees. An error in VNODE cannot possibly be ruled out although it was 67 Table 2. Accuracy and speed of convergence for methods of solution. Element Press. Flow Lateral Comp. Number Method Type Unif. Unif. flow time of [gpm] [sec] iter. Exact 88 94 3.43 70.13 15 Method Linear 88 94 3.45 2.19 8 No 1 Quad. 88 94 3.45 1.42 8 Point Method Linear 88 94 3.45 2.63 9 Sources 2 Quad. 88 94 3.46 1.92 9 Method Linear 88 94 3.45 2.19 8 3 Quad. 88 94 3.46 1.92 8 Method Linear 88 94 3.45 12.02 8 Point 1 Quad. 88 94 3.45 26.52 8 Sources Method Linear 81 91 3.21 16.2 10 2 Quad. 88 94 3.45 36.62 . 9 Method Linear 85 93 3.33 17.9 9 3 Quad. 88 94 3.45 32.02 8 checked into. For most methods it is though that these deviations in flow rates are insignificant compared to the variation due to manufacturing most emrniters. One could probably calculate a "estimation uniformity" that would decrease the hydraulic uniformity according to this error estimation. 69 A8988 ~53 o: v dean“ .325 eve: h .m> 3.53% 3580 63m AN 0.8mm..— n top-.02 d N Von-02 O — $0503 + .036 .5. too... 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Ease has. can a 2&2... n .55.: c u .55.: o — 3.3.: + .36 :5 one... 2:38.. .096 ON wN .vN NN cN _ — p _ b .- h — ON on Anemone; Cot—Eng .coEoB vacate-.0 c0520m .5 eotm [1.1.11 «ruddy poor-r IJHIIIJ‘ 72 A8988 “Eon v Road“ cognac coo: h .m> ”Banana .8qu Saw .3 Bani n .35.: a n .35.: o . .55.: + :5 veo: 8:39... .036 on 0N on ow «N p e _ _ _ _ e _ _ Anger-Eu oflaem .Ev .coEofi 9:95.50 cozqzom .6 not“... .uoxu ON on 9‘ s [w] 'xeaddv poo... ”none.“ 73 4.2.2 Initial Estimates and Convergence The sensitivity of initial estimates of pressure head at each node varied greatly depending on which solution method was used. Method 1, suffered from the fact that initial estimates at each node had to be fairly accurate in order for the solution to converge. For example, if a linear element is chosen and initial estimates at both nodes are the same then Ali/Ax for the element is zero, and a zero results on the diagonal of the solution matrix causing the solution to stop. In addition, other initial estimates can be chosen to cause the solution matrix to have off diagonal terms causing the solution to diverge. Methods 2 and 3 didn’t suffer from this problem. This is due to the additional G(h) term which adds directly to the stiffness matrix of setting any terms that ended up negative or zero when [K1,] was added to the solution matrix. The solution was considered arrived at when the maximum defect each iteration reached a predetermined value. For the system shown in Figure 19 a plot of the maximum defect from each iteration is shown in Figure 25. All methods converged to a relative accuracy after a about 8 iterations to a maximum defect of 0.01 for the system in Figure 19. 74 M— 883 8:832: .m> 53.8: .3 95mm c0393: N— : o— fiT IT hi l —.o 1 n6 1 v.0 r. n6 c0393: .9 .uouoo Edi—x02 co::_om *0 02.60.5250 rootoa tummxen 75 4.3 Summary Discussion The results and discussion section have shown the advantages and disadvantages of using a virtual node concept in analyzing hydraulics of a drip irrigation system. One of the advantages of using the concepts demonstrated in this thesis are the speed with which an analysis can be carried out. By combining emitters into virtual nodes the computation speed with which an analysis can be carried out is increased tenfold. Large systems of 10,000 or more emitters can be analyzed more accurately by using a virtual node system. A very important limitation of the virtual emitter concept is the element should have the same pipe diameter and flow characteristics throughout for the method to be used sucessfully regardless of the number of emitters that are combined into virtual nodes. If the pipe diameter changes in the middle of the element the element should be divided into two elements at the point the change takes place and seperate flow characteristics of each element can be accounted for in the system of equations. By incorporating emitter flow as described in the theoretical development, good initial estimates are not required to carry out the computer analysis and research shows that convergence is almost always guaranteed. It was shown that the flow phenomena occuring in a drip irrigation lateral could be described by a differential equation [38]. The solution of the differential equation of this form is readily solved using classical finite element techniques. Using the concepts developed in this thesis, a large drip irrigation submain unit can be discretized into a convenient number of linear elements of any type and analyzed using a general finite element program. 76 V. CONCLUSIONS AND RECOMMENDATIONS In this thesis, the use of virtual node concepts were investigated in the hydraulic design of drip irrigation systems. A differential equation was developed to describe the .flow conditions in a drip lateral. This equation was solved using the finite element method and virtual node concepts in the theoretical development section. A computer program, VNODE, was written to evaluate these concepts. This resulted in a fast, efficient and accurate solution in most cases. The specific conclusions of this research are: 1. Virtual node concepts and the finite element method can be used to analyze the hydraulics of a drip irrigation system. 2. Combining emitters into virtual nodes resulted in less computation time and approximated the exact solution closely. 3. Using a virtual node system, a good analysis can be performed without the need to directly solve for pressures at every emitter. 5‘ Using the differential equation deve10ped, quadratic elements can be used to analyze 77 the flow in a drip irrigation lateral. Recommendations for further research 1. In the theoretical development it was shown how slope or elevational effects could be incorporated into the solution, these effects could be included for further study. 2. To make this an effective design tool for analyzing hydraulics of drip systems VNODE could be incorporated into a more easily used program. 0 3. Large drip irrigation submain units could be analyzed using these methods using a general finite element program. APPENDIX 78 APPENDIX A Computer Model (VNODE) PrograrnListing program vnode; uses Crt. Printer, Dos; const MaxSize a 65; Xe a 0.5; Ke = 9.14E-7; C a 150; pi = 3.14159; type MatrixType = array [1. .MaxSize,l ..MaxSizc] of real; Vector = array [1..MaxSize] of real; var k.Kg : Matrix'l‘ype; oldh ,:h,z Vecta'; materN.1J.Ne.r.8.t.u in er. Dx ,G ,Q,Mdbc ,debc,epl,l ,DiaNodeFlows : Vector; ltype: arrayIl. .MaxSize] of integer; PsmPsxPst, Y: Vector, Nel: array[1..MaxSize,l..4] of mteger; outdata,plotfile: text; L ,:Totall-'-'low real; filename: string[l4]; w,Nf 1n ’Ngn: gyl Infilaxsize] of rnteger; defecLGtemp ,zreathemp hour,rnin,sec,hndth: word; function Power(Base.Pwr : Real) : Real; [ FunctionPowerraisesBasetothepoweerr. 'n { Power ) if Base>0.0 then lsePower:=crrp(l’wr‘l.n(Base)) e Power:=0.0; end: 1 Power] 79 function Length(p : integer) : real; var ij : integer; begin i:=Nel[ ,1]; typepjlalthen :=Ne1[p .2] else j .=Nel[p .3]; mlagnsth: a2611116qutil-XLi])+sc1r(Ylnl- YUD); ftmction LinearNi(e : integer, Xo : real):rea1; var i = integer. begin j. '];=Nel[e.2 mLhtearNirsm [jl-XOM-GISWC). function LinearNj(e: integer; Xo : rcal):real; var i: integer: i::-Ne1[e,l]; “ENNitsao-Xlilmwmme); function we : integer; Xo : rea1):rea1; var i :. integer; i:-Nel[e.l]; and911819330-2"'(X0-X[i])/L€fl81h(¢))"'(1(KO-X[i])/L¢l|81h(¢)): function Quade(e : integer; Xo : rea1):real; var i: integer; begin Ne1[e, i: a: mmj: =(4’1X0-X1i])/lmsfll(e))*(l-(Xo-X[il)/Lalsth(e)): function We : integer; Xo : real):rea1; var i : integer: begin i:=Nel[e,1]; ”8W4!‘(Xo-Xli])llmmh(e))‘(l-2"(X0-X[i])/Lensth(c)): 80 function DxCalc(i,i,m:integer):real; var baseaXArea : real; WW‘WINW: 3856: =ab6((h[il-h[i])/Lenglh(m)): a:= 5. 88/(power(C, l .852)* po.wer(XArea,0 5835)); if(baso:0. 0) or (a==0. 0) then poI).rtCalc DxCalc: =(XArea)/(power(base, 0 .46)‘power(a,0.54)); end: function Gcalc(rn.Ps:integer):real; var base : real; Lil: : integer: i :3 Nellm,l]; j :- Nellm.2]: k :- Nel[m,3]; ifItypeIrn] = 1 then base :- LinearNi(rn.Psx[Ps])*h[i]+LinearNj(m,Psx[Ps])"'h[i] else a QuadNi(m.Psx[PSD"'h[i]+Quade(m.PsleS])*hLi]+Quade(m.Psx{P81)"h[k]: { GCalc:=Xe*Ke'powet(baserx°-1);l { GCalc: =Ke‘power(base.Xe-l); )] function QCalc(m,Ps:integer):real; var base: real; iii): : integer. begin i. '8 Nellm, l]; i = Nellmzl: I Nel[m,3]; if ltypelm] . 1 then gage: sLinearNi(rn,Psx[Ps])"h[i]+LinearNj(m,Psx[Ps])"hU] e P8131361;;flawQIIIKINKIIIJ’rIx[l’-'>l)"'h[il-I-Qnadl‘lflm.PSXIPSI)"II1.i]+QIIIIdI‘1k(m.Psx[1’8])"‘h[k]: { QCalc: s-l"(l-Xe)*l(e"power(bme,Xe);] QCalc :=-l‘l(e‘power(base X6); 1 QCalc:-0: } end: 81 function DxnCa1c(hl,h2:reaI;m:integer):real; var base,a : real; begin writeln(’"“’ [1n] :4,hl: >103. h2: 10: 3.1.en (m 10: 3); Base: =abs((hl-ht2pe/Length gth )' a:= 10.59] wer(C, l .852)(*power(Dia[m], 4 .87 1)); if(basa=0. )or(a=0.0) then DxnCalc:=0 DxnCalc:= l/(power(base, 0 ..46)*power(a,0 54)); end: function GnCalc(hl,h2:real;rn:integer):real; var base : real; begin base :s abs((h l+h2)/2); { GnCalc:=Xe*lp1[m]‘epl[m]*l(e*powa(base x91); 1 { Eggpummmm'm‘wweflbaRXe-1);1 c: , end: function QnCalc(hl.h2:rea1;m:integer):real; var base : real; 1 fiafliiiizi’ixhu ] ll lKe‘ WK):1 c- - - e "‘ m ‘ep 1n * po 6 QnCalc. :=-l'lpl[m]*epl[m]*Ke*nower(base,Xe e;) { QnCalcz-O; end: procedmeSolve(vara:MatrixT‘ype; varx : Vectonn : integer); 14.1: : Imam prvocmult : real; begin {forwardprocedmel for k:=1 to 11 do begin Pivot -= m1; for j.- =k to n+1 do alk,j]. = a[kJ]/pivot; for 1. =k-1-l to 11 do begin mult := stilt]: 82 for jzak to n+1 do aIiJ] := alij] - mult‘alkJ]: end; 1 backward pmcedm l for i:=n downto 1 do begin for jzsl to 11 do x[i] := x[i] - x[jl'aIiJ]; 3191]; := x[i] + a[i,n+l] 9 end: procedure MatShow(var MatrixA: Matrix'l‘ype; Rows, Columns: integer); var I, J: integer, begin for 1:31 to Rows do for szl to Columns do .mgMamxAHlelo): Wfl old: wnteln; end: procedure DSPLinearlD(rn : integer); 1 :mteger; lireal: i:=Nel[m,l]; i==Nel[m.21: L==L¢fl8lh(In): Kliilz=K[i.i]+DXIml/L: Kljj]:=l([!,j]+Dx[m]/L; Klgdlt=Klgdl-Dx{ml/L; Khalz=KD.Il-Dx{ml/L: Kliilz=K1§JI+G[m]‘U3: KDJII'KDJJ‘l'Glml'Ifi: Klgdlz=Klgdl+G[m]*U6; Khalz-KUJHGImPUG: K[i,N+1]:sK[i,N+l]+Q[m]‘L/2; KUN+ll:-KU.N+11+Q[ml‘m: Kg[i,i]:=Kg[i,i]+G[m]*Lf3; . . . . Kgfia]:=Kg[i.j]+G[m]*L/3; { The [Kg] matrix is saved so 1t1s easrer to ] Kg[iJ]:=Kg[i,j]+G[m]*L/6; { calculate nodal flows later on 1 Ksfidlt-Ksbalmlml‘w; 83 end: procedure DSPQuad(m : integer): type Coeff = arrayll..3,l..3] of integer; GODS! CK : Coeff = ((4.2.-1).(2.16.2).(-1.2.4)); 2 CM 8 ((l4o°16t2)9('169329.16)9(2o'16914)); var r,s,t,i,j : integer; Lzreal; begin r:=Nel[n1,l]; s:=Nel[rn,2]; 131%! [111.3]: L==1wzth(m; KIIN+llt=Ker+ll+lel’l/6: K[s,N+l]:=K[s,N+l]-14*Q[m]*L/6; KIW+112=KIIN+11+lel‘L/6: fm' i:=1 to 3 do for j:=l to 3 do . . KlNcllmJJNellmdllt=KlN¢l[mi].NellmflkGlml‘L'CKBllJIBO; for i:=l to 3 do for j:=l to 3 do . . KlNellmJJNellmjll:=-=K1Nel[m.il.NcllmilkDXImll6/L‘CKDXDdl: for i:=1 to 3 do for jzsl to 3 do . KgINellm.il.Ncl[m.illz-KslNel[mimcllmjlkGlmPI-‘CKSIIJTBG. end: procedure PsLinear(rn : integer): var Qstnr,Gstar,Ni,N j : real; ij: mteger; i:-Nel[Pse[m],l]; irNleselmlll; Qstm==Psrn[m]‘QcaIC(Pse[m].m); Gstar:=Psm[m]*Gcalc(Pse[m],rn); Ni:=LinearNi(Pse[m],Psx[m]); Nj:=LinearNi(Pse[n1],Psx[m]); KliN-tlleliNHIWNI: KUN+11==KUN+11+QWNh Ktiriz=Kti.ii+Gstar*sqr(N9; K[j,j]:=K[jj]+Gstar‘sqr(N]); K[1,j]:=K[1J]+Gstar‘Nj*Nr; KUJ]:=KU.i]+Gstar‘Nj‘Ni; Kaliilz=Ksliil+Gstafisqr=0.01) and (iter<=20)) do begin for i:=l to Ne do begin [i121 thenbe fiin Ili=DxCalc{Ne[i.ll.Nel[i.21.i); if It 1]=2 then Dfimwcmefiinrvdhsm end; for i:sl to N do for j:=l to N-t-l do K[i,j]:=0; for izsl to N do for j:I-l to N+l do Kg[i,fl:-0; if Ps-l then for i:=l to Nps do if ItypelPseli]]=l then PsLinear(i) else PsQuad(i); if Ps=0 then be for i:=l to Ne gbegin if It ype[i]=l then begin Gli]. m=6n€a1c(h[Nclli.lll.thd[i.2]l iii) <32]:— .=QnCaIC(h[Nel[i.lll.h[Nel[i.2]1.i); if It [i]=2 then begin GI I==GnCal¢(thel[i.lll.h1Nel[i.3]l.D: QliltsQnCalcmlNellulllthellulld): 87 end; end; end; for i:=l to Ne do if Itypeli]=l then DSPLinearlD(i) else DSPQuad(i): for i:=1 to N do Nodellowsli]:=K[i.N+l]; for i:=l to Nan do Modify(an[i]); {Matshow(k,n,n+ l);for has 1 to Ne do writeln(Dx[i]:20:15);] Solve(K.h.n): defect:=0; for i:=1 to N do if (abs(h[i]-oldh[i]) >8 defect) then defect:=abs(h[i]-oldh[i]); for i:=1 to N do oldh[i]:=h[i]; writeln(iter:4,defect:8:4); iter:-itcr+l; end; GetTWMsechnddI): writeln(out); writeln(out,’ Calculation Time [sec] ’rnin‘éO-I-sec-I-hndth/lOOflS); writeln(out); writeln(out,’ Number of Iterations’itert4); writeln(out); writeln(out,’ Maximum defect ’,defect:7:5); writeln(out); writeln(out,’F'mal Coefficients Dx, G, and Q for each element’); writeln(out); for m: =1 to Ne do begin i:=Nel[m, 1]; J =Nellm2]; writeln(ouLDxIm]: 15: 10 ,:G[m] 15:10,:le] 15:10); end; for'1:= l to N do for j: =1 to N do NodeFlowsli]: =NodeFlows[i]- -Kg[i,j]"h[i]; writeln(out); writeln(out.’ Final Nodal Pressures NodalFlows {F}+[Kg]* {h} ’); writeln(out); writeln(out,’ meters feet m3/sec gprn’); writeln(ouL’Node’); for 1:= l to N do begin writeln(out,i: 3,h[1]: 10:2 ,(3 .281‘h[i]): 20:2, ’ ’ NodeFlowsli]: 15: 8 (1.5835 31"NodeFlowinl): 15: 2); end: writeln(out); TotalP‘low:=0; for i:=l to N do TotalFlow:=TotalFlow+NodeFlows[i]; writeln(out,’ Total Submain Flow ’,'1‘ota1Flow: 15:8, (T'otalFlow‘15835.3 1): 15:2); writeln(out); writeln(out,’ ""“" 1D Linear Elements """’); writeln(out); writeln(out.’ Element flows, Dx‘Hf/L Average Emitter Flows, Qle]+G[e]*h ’); 88 writeln(out); writeln(out,’ m3/scc gpm m3/sec gph’); writeln(ouL’Element’): for m:=l to Ne do begin if Itypelm]=l then begin i:=Nel[m,l]; i:=Nel[m.2]: minimum:5.(Dx{m}*(h[il-hU1)/Lenglh(m))= 15:8. (15835.31*Dx[m]"'(h[i]-hU])/Length(m)):12:2); if (epl[m]=0) or (1pl[m]=0) then writeln(out,epl[m]:l7:8,epl[m]: 17:2) else Wfiteln(0uI.(Q[m]-G[m]*(hli]+hLi])/2)/cpllmlflpllmlt17:8. end. (15835.3l*60"(Q[m]-G[m]‘(h[I]+hU])/2)/epl[ml/lpl[ml):17:2); end. writeln(out); writeln(ouLNpsz4,’ Point Sinks (emitters)’); writeln(out); if Pg] then writeln(ouL’Emitters considered as point sinks’) else writeln(ortL’Emitter flow averaged over the element’); writeln(out); writeln(out.’ Element X Pres.[m] Outflow [gph]’); writeln(out); for i:=1 to Nps do begin . . Gtempz=Gcalc(Pse[1],1); QT emv =QcaIC(Psc{il.i): Bflow: =PSIn1i]"(GtcmP"Psy1i l-QIeIIIp): writeln(ouLPseli]: 10,Psx[i]: 12' 2.Psy[1]: 12: 2, 15835.31‘60‘Eflow: 12:2,Psm[i]: 12:2); Close(dut); {Plotiu} end. 89 APPENDIX B Y Headh (m) EICVZ (m) ijkm Dr: G Q epllpl Dia Sample Output 7 Nodes mmmmmm 000000 mmmmmm 0.0.0.0.0.0. 234567 123456 Y Modifier 30.1!) X Element Head, h (m) 51 Point Sinks (emitters) 1 Known Nodal Values Element mmmmmmmmmmmmmmmmmmm .......... Lllllllllllllllllll mmmmmmmmmmmmmmmmmmm 0.000000000000000000 wmmmmmmmmmmmmmmmmmm iwmmfiwfi fiwfiwfimfi um” 1111111122222222333 3 100.00 0.00 1.00 3 105.00 0.00 1.00 3 1 10.00 0.00 1.00 3 1 15.00 0.00 1.00 3 120.00 0.00 1.00 3 125.00 0.00 0.50 4 125.00 0.00 0.50 4 130.00 0.00 1.00 4 135.“) 0.00 1.00 4 140.00 0.00 1.00 4 145.00 01X) 1.00 4 150.00 0.00 1.00 4 155.00 0.“) 1.00 4 160.00 0.00 1.00 4 165.00 0.00 1.“) 5 170.00 0.00 11!) 5 175.00 0.00 1.00 5 180.00 0.00 1.00 5 185.00 0.00 1.00 5 190.00 0.00 1.00 5 195.00 0.00 1.00 5 71X).00 0.00 1.00 5 205.00 0.00 1.00 6 210.1!) 0.00 1.00 6 215.00 0.00 1.00 6 220.00 0.00 1.00 6 225.00 0.00 1.00 6 230.“) 0.00 1.00 6 235.“) 0.00 1.00 6 240.00 0.00 1.00 6 245.00 0.“) 1.00 6 250.00 0.00 11X) Calculation Time [sec] 2.190 Number of Iterations 8 Maximum defect 0.00453 Final Coefficients Dx, G. and Q for each element 0. 0&0426272 0 .0000000000 -0. (”00009670 0. 0024604603 0.0000(X10000 011100009070 0. 0030791205 0.0000([10000 -0.(X)00008661 0.0041248804 0.0000(X)0000 -0.(X)00008413 0.0063850525 o.oooooooooo 011100008292 0.0162336773 o.oooooooooo 00300008253 Fmal Nodal Ptecaues NodalFlows [F}+[l(g]* {h} meters feet m3/sec gpm Node 1 30.00 98.43 000002015 032 2 25.96 85.18 000003904 062 3 23.27 76.34 000003694 058 4 21.62 70.93 000003557 056 5 20.74 68.06 000003480 055 6 20.41 66.95 000003447 055 7 20.36 66.81 000001720 027 Total Submain Flow 00m21816 -3.45 91 0“... 1D LinearElements tit... Element flows, Dx‘Hf/L Average Emitter Flows. 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' '."'.... 000 0.00“ 00“ 000 0.000 0'“ 000 0‘00“ 0'“ 000 0‘00“ 00“ 000 0u000 000 000 '¢00N IN“ 000 0¢00N 000 000 5:00“ 0¢N 000 0'00“ 0¢N 000 «0¢N 00“ .000 0000“ 00“ “000 0000“ 00- 0000 000R 00. N000 '00ON 00- N000 00000 00- 0000 0'00“ 050 0000 «000“ 05- '000 0000“ 00- 0000 00000 00- 0000 0500“ 00¢ 0000 '0000 00. 5000 00000 0'¢ 0000 000.0 0'¢ ' 000‘“ 00a «00' 000.0 00- N00' 5'0-0 IN. '00' N00.“ 00. 000' 050-“ 0a. 500' 500'“ 0¢¢ 000' 50000 00. ¢¢0' 500a“ 00. 0¢0' 00000 00 0¢0' 00000 00 500' 50008 00 «0' '000N 00 NN0' 0000“ 05 0N0' «000“ 05 5N0' N00'“ 00 00' 000'“ 00 000' 000'“ 00 000' 08000 00 '0' '000N 0' 0'0' '0000 0' 5'0' 0'00“ 00 00' 000 00 '00' 0005“ 00 000' '0050 00 N00' 0000“ In 500' 50000 Ga «50' “'00“ 0 00 0 08500 SOuL 080 ‘00‘ —80 LOUUuIU OL’IQOLS 50 J 00008 00 LIST OF REFERENCES 98 LIST OF REFERENCES Barlow, LP. and E. Markland. 1969. Computer Analysis of Pipe Networlts. Proceedings I.C.E. 43:249-259. Benedict, RP. 1980. Fundamentals of Pipe Flow, John Wiley and Sons. 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