‘r‘lw‘ {I‘JJ 1", ',‘\.' ‘4 ‘2 «an 3,...- MSU LIBRARIES RETURNING MATERIALS: P1ace in book drop to remove this checkout fr.~ your record. f1!§§_wi1‘ be charged if book is returned after the daf' stamped below. ..- a,“ -- O HYDRAULIC DESIGN OF SPRINKLER IRRIGATION SYSTEM COMPONENTS USING THE FINITE ELEMENT METHOD BY LUIS ALFONSO SALDIVIA 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 1988 //. 7‘. 5-57.-” {.4- Efi)/ /Zx. /( ABSTRACT HYDRAULIC DESIGN OF SPRINKLER IRRIGATION SYSTEM COMPONENTS USING THE FINITE ELEMENT METHOD BY LUIS ALFONSO SALDIVIA The objective of irrigation has been to increase both the quality and quantity of food production by allowing the timely delivery of water to meet crop requirements. To attain maximum utilization of the existing water resources it is necessary to employ irrigation water in the most efficient manner. Proper hydraulic design of irrigation system considering the system components such as elbows, tees, gate valves, sprinklers, and pumps has become the key to irrigation efficiency. The primary objective of this research was to incorporate the system components in the design of sprinkler irrigation networks. The procedure was based on the solution of the nonlinear algebraic equations derived from the finite element formulation of the system components. The secondary research objective was to assess the applicability of the finite element formulation employed in the design of sprinkler irrigation systems by comparing it to a linear theory model. ACKNOWLEDGEMENT The author wishes to sincerely thank Dr. Larry Segerlind, Dr. David Wiggert and Dr. George Merva for their teaching and guidance. To my major professor Dr. Vincent Bralts for his never ending encouragement and support through this learning experience. To my friends Rabi Mohtar, Shaun Kelly, Walid Shayya, Sally Wallace and Kevin Rose with whom I have shared the enjoyment and rewards of graduate school. Finally, to my parents and most of all to my dear wife Beth for all the support they have given me. ii TABLE OF CONTENTS LIST OF TABLESOOOOOO0.0.00.0...OOOOOOOOOOOOOOOOOOOOOOOv LIST LIST I. II. III. IV. OF FIGURESOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOVi OF SYMBOLS...OO0.0.0.000...OOOOOOOOOOOOOOOOIOOOOOix INTRODUCTION.OOOOOOOOOOOOOOOOOOOOOOOOOOOO...00.0.1 A. Scope and Objectives..........................5 REVIEW OF LITERATURE AND THEORY..................8 A. Basic Hydraulics of Sprinkler Irrigation......8 B. Methods for the Analysis of Pipe Networks....29 C. Analysis of Hydraulic Components.............52 D. Summary....... ..... ..........................62 METHomLOGYooooooooooooooooo00000000000000.0000063 A. ResearCh ApproaChOOOOOOO00.0.000000000000000063 B. Theoretical DevelopmentOO0.0.0.0000000000000065 RESULTS AND DISCUSSION..........................79 A. Computer Models..............................79 8. Model Comparisons............................80 C. Accuracy and Reliability....................ll3 D. summarYOOOOOOOOOOOOOO0.0.00.00.00.0000000000114 V. CONCLUSIONS AND RECOMMENDATIONS................116 APPENDIX APPENDIX APPENDIX APPENDIX Taylor Finite Linear Models Series Expansion...............118 Element Computer Program.......120 Theory Computer Program........135 Simulation Data0000000000000000138 LIST OF REFERENCESO..00..OC0.00000000000000000000000.162 iv LIST OF TABLES Table 1. Loss coefficient for common components.......... 56 2. Results of the correlation and linear regression analysis for the finite element versus the linear theory model for junction heads..................87 3. Results of the correlation and linear regression analysis for the finite element versus the linear theory model for sprinkler flows.................88 4. Results of junction head comparisons between the finite element and linear theory models..........97 5. Results of total sprinkler flow comparison between the finite element and linear theory models......99 6. Results of the correlation and linear regression analysis for the finite element versus the linear theory model for junction heads with a pump.....104 7. Results of the correlation and linear regression analysis for the finite element versus the linear theory model for sprinkler flows with a pump....104 8. Results of the junction head comparisons between the finite element and linear theory models with a pumPOOOOOOOOOOOOOOOOO.0...00.0.000000000000000110 9. Results of the total sprinkler flow comparison between the finite element and the linear theory mOdels WithapumPOOOOOOOOO0.0.0.000000000000000110 Page LIST OF FIGURES Figure Page 1. Sprinkler irrigation system and components........4 2. Pipe section.....................................1l 3. Relationship between F-factor and number of sprinklers on a lateral line for cases I and II..18 4. Lateral line notation and dimensionless gradient line variables...................................21 5. Pressure distribution for downslope conditions...25 6. Pressure distribution for upslope conditions.....26 7. The Newton-Raphson method for the one dimensional case.................................35 8. Sequence of nodes and elements...................47 9.‘ Straight pipe element section....................49 10. Sprinkler irrigation components..................53 11. Pipe section and system components...............66 12. Sequence of nodes and elements...................73 13. Example sprinkler irrigation system..............76 14. Example solution matrix..........................78 15. Sprinkler irrigation system layout...............82 16. Example finite element computer program output...84 17. Example linear theory computer program output....85 18. Comparison of finite element and linear theory solution for junction heads with 0 percent slope and KC equal to 0.0 0.00000000000000000000...0.0.89 vi 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. Comparison of finite element and linear theory solution for junction heads with 1 percent upslope and Kc equal to l.0......................90 Comparison of finite element and linear theory solution for junction heads with 2 percent downslope and Kc equal to 2.0....................91 Comparison of finite element and linear theory solution for junction heads with 2 percent upslope and Kc equal to 3.0......................92 Comparison of finite element and linear theory solution for sprinkler flows with 0 percent slope and KC equal to OCOIOCOOOOOOOOOOOO0.0.0.00000000093 Comparison of finite element and linear theory solution for sprinkler flows with 1 percent upslope and Kc equal to 1.0......................94 Comparison of finite element and linear theory solution for sprinkler flows with 2 percent downslope and Kc equal to 2.0....................9S Comparison of finite element and linear theory solution for sprinkler flows with 2 percent upslope and Kc equal to 3.0......................96 Hydraulic grade line representation for 2 percent uphill and Kc equal to 3.0..............100 Hydraulic grade line representation for 2 percent lepe downhill and Kc equal to 2.0......101 Sprinkler irrigation system layout with a pump..103 Comparison of finite element and linear theory solution for junction heads with 1 percent upslope and Kc equal to 1.0 with a pump in the system......................................105 Comparison of finite element and linear theory solution for junction heads with 2 percent downslope and Kc equal to 2.0 with a pump in the system......................................106 Comparison of finite element and linear theory solution for sprinkler flows with 1 percent upslope and Kc equal to 1.0 with a pump in the system......................................107 vii 32. 33. 34. Comparison of finite element and linear theory solution for sprinkler flows with 2 percent downlepe and Kc equal to 2.0 with a pump in the system......................................108 Hydraulic grade line representation for 2 percent downslope with Kc equal to 2.0 with apump in the system000000..0...0.............00111 Hydraulic grade line representation for 2 percent upslope with Kc equal to 3.0 with a pump in the system.0......0............O........0.0.0.00112 viii Ax(n) dz AHS AQ(n+1) AQ(n) LIST OF SYMBOLS a constant in the friction loss equation total cross-sectional area of the sprinkler nozzel pump coefficient pipe roughness coefficient linearized pipe constant linearized component constant linearized pump constant linearized sprinkler constant sprinkler discharge coefficient pressure distribution with respect to length slope of the energy line correction to the x variable slope of the ground pipe diameter differential equation constant inverse of the Jacobian matrix, D tolerance value total friction loss friction loss in the lateral line friction loss in the submain vector of newly extimated corrective flows vector of previously estimated corrective flows ix {f} f(8) absolute value of the mean deviation of irrigation depth dimensionless friction factor force vector element contribution to the force vector vector of the function of x friction loss correction factor acceleration of gravity specific weight of water density of water component head loss head loss head for a given length ratio total energy (P/y + 2) maximum lateral line pressure minimum lateral line pressure lateral line pressure variation pump head cut off head pressure drop at length ratio i fixed grade node junction pressure at the upstream node junction pressure at the downstream node length ratio R/L constant of proportionality for the sprinkler discharge equation element contribution to the stiffness matrix stiffness matrix {0} Qi go AQ(n) Q(i-1) Q(i-2) R“) RiAH RiAH' pipe line constant which is a function of line length, diameter and pipe material modified pipe line constant component coefficient sprinkler constant length of pipe length of pipe measured at desired location empirical head loss exponent pump coefficient number of sprinklers iteration number shape function ratio of the circumference of a circle to its diameter static pressure head unknown pressure vector of unknown pressures sprinkler discharge discharge at a given length ratio i original discharge flow rate flow correction flow rate from the previous trial flow rate from the trial previous to Q(n-l) vector of residual equations energy drop ratio head loss due to friction head loss or gain due to slope xi x(n+1) x(n) z(n) energy line Christiansen uniformity coefficient statistical uniformity coefficient velocity velocity head coefficient of variation for the depth of irrigation water weighting function friction loss, including minor losses as equivalent length vector of new extimates of variable, x vector of first estimates of variable, x mean depth of irrigation elevation head solution vector of the linear system xii I . I NTRODUCT I ON Irrigation technology has been known and practiced for at least 5,000 years. It is an important agricultu- ral technology, one that has facilitated the development of many civilizations, particularly in arid regions. The primary benefits of irrigation have been to increase both the quantity and quality of food production by allowing the timely delivery of water to meet crop requirements. It is estimated that the total irrigated area in the world has increased from 95 million hectares in 1950 to approximately 250 million hectares in 1986 (Power, 1986). While irrigated areas only comprise 20 percent of all harvested land, it accounts for 40 percent of total crop output worldwide. Many different irrigation technologies are employed throughout the world. These include surface, subsurface, sprinkler and drip irrigation. Surface irrigation is the most commonly practiced irrigation method worldwide. However, since World War II, more efficient forms of irrigation such as drip and sprinkler irrigation have increased in popularity. This is primarily due to the introduction of aluminum, galvanized steel and plastic pipes in EurOpe and the United States. Sprinkler irriga- tion and, to a greater extent, drip irrigation apply water in very controlled and precise amounts. Therefore, these methods ultimately provide a higher water use efficiency and greater uniformity than surface irrigation. In the United States sprinkler and drip irrigation have expanded to 9 million hectares in the last two decades and presently account for over 35 percent of the irrigated lands (Irrigation Journal, 1986). In Michigan, sprinkler and drip irrigation account for over 95 percent of the land under irrigation or approximately 175,000 hectares (Irrigation Journal, 1986). Although Michigan is considered quite humid, the majority of the measured precipitation occurs with winter snow. Short term summer droughts are common, thus irrigation can eliminate the most costly and unpredictable farming hazard; insufficient moisture at critical growth periods. In addition, sprinkler irrigation systems in humid areas such as Michigan have been designed and operated for frost and freeze protection, blossom delay, and in some cases crop cooling (Jensen, 1980). A common sprinkler irrigation arrangement is a solid set system. These systems may be installed on the surface of the ground using portable aluminum pipe and are usually left in place for the entire growing season. In a solid set system as shown in Figure l, the water flows sequentially through the following components: the water supply and pump set, main line, lateral lines, riser assemblies, and sprinkler heads. In addition, elbows, tees, gate valves, and pressure regulating valves form part of the system. Because expanding needs of industry along with increasing population are making great demands on the limited water supply, conservation of agricultural water resources has become a worldwide priority. To attain maximum utilization of the existing water resources it is necessary to employ irrigation water in the most efficient manner. Proper design of irrigation systems considering pressure losses through irrigation components such as elbows, tees, gate valves, and sprinklers, as well as along laterals, is the key to irrigation efficiency. A. SCOPE AND OBJECTIVES The ultimate purpose of irrigation design is to determine the optimal size and arrangement of the system components (pipes, pumps, nozzles, etc.) so that the crop water requirements can be met without violating constraints on water, energy, and investment. Sprinkler and drip irrigation system design procedures have been presented by several researchers: Lateral Line Suction Line Submain Line .’/ I Pump < lflg 4/ Motor / Figure l. Sprinkler irrigation system and components (Schwab et al., 1981) Howell and Hiler, 1974; Keller and Karmeli, 1974; Wu and Gitlin, 1974, 1983; Pair, 1975; Perold, 1977; Wood, 1980; Jensen, 1982; and Bralts and Segerlind, 1985. Lateral length and diameter, orifice spacing, slope, pressure, flow rate, and uniformity are important parameters which have been considered in these design processes. Until now, in considering pressure losses from the water source to the outlet, only the losses occurring in the sprinkler or drip lateral were considered important. Wood (1980), Howell and Barinas (1980), Finkel (1982) and Haghighi et al. (1987), suggested that, in addition to pressure losses in laterals, the pressure losses across hydraulic components such as elbows, tees and valves in a network of pipes should be considered in the system design procedure. In most of the literature the pressure losses across hydraulic components have been termed "minor losses", but since a large number of these components may be present in a single sprinkler irrigation system, these minor losses can become significant and alter considerably the hydraulic design. Thus, a pronounced need has arisen for the inclusion of tees, elbows and valves in the irrigation design process. The design of sprinkler irrigation systems is an intensive and complicated procedure which involves a large number of calculations. The use of microcomputer based iterative techniques removes much of the tedious work associated with these calculations. This results in a more detailed analysis with fewer errors when compared with conventional techniques. The use of microcomputer based iterative techniques in the design of network irrigation systems has been growing steadily. Edwards and Spencer (1972) presented a computer based design procedure for the analysis of sprinkler irrigation systems utilizing the Hardy Cross method. Other investigators that have employed iterative techniques to solve for flow rates and pressures in drip irrigation systems, based on assumed end line pressures, were Solomon and Keller (1974), Wu and Fangmeier (1974), Howell and Hiler (1974), and Perold (1977). Wood (1980) included sprinkler irrigation design in his hydraulic network analysis using linear theory method. Bralts and Segerlind (1985) developed an interactive microcomputer program (DESIGNER) which uses the finite element method to assist in the analysis and design of drip irrigation submain units. DESIGNER is based on the iterative solution of a set of linearized flow equations and the result is a symmetric banded matrix which requires minimal computer storage. At this point in time the DESIGNER program only evaluates pressure losses from lateral lines and does not include pumps and other essential components of sprinkler irrigation systems. The purpose of this study is to incorporate system components such as tees, elbows, gate valves, and pumps in the design of sprinkler irrigation systems. The procedure is based on the solution of nonlinear algebraic equations employing the finite element method. The specific objectives of this research are : 1. To develop a finite element formulation of essential sprinkler irrigation components such as elbows, tees, gate valves, sprinklers, and pumps. 2. To demonstrate the potential of applying the finite element formulation to the design of sprinkler irrigation systems by comparing it to a linear theory model. I I . REVI EW OF LITERATURE AND THEORY In order to understand the vadility of the proposed design analysis it is necessary to present a review of important literature and theory. The topics to be developed in this section are a) basic hydraulics of sprinkler irrigation, b) methods for the analysis of pipe networks, and c) analysis of hydraulic components. A. Basic Hydraulics of Sprinkler Irrigation A sprinkler irrigation system is composed of a series of pipes sections and of a number of hydraulic components such as tees, elbows, valves, and pumps. The pipes convey the water through the various components and supply it to the sprinklers at the correct pressure head. A diagram of a sprinkler irrigation system is presented in Figure 1. An important element in sprinkler irrigation design procedure is the determination of pressure losses occurring in the sprinkler lateral. This pressure loss occurs when a fluid is forced through a pipe section increasing friction and causing energy to be consumed. This results in a drOp in pressure across the pipe section. 1. Governing Equations The analysis of fluid flow through networks of pipes is based on Newton's laws of conservation of mass and energy. Conservation of mass or Continuity states: 3Q = 0 [l] or that the flow in a network system must be balanced at every junction (node). In essence, the flow into a node must equal the flow out. Conservation of energy states: 2H = o [2] or that the algebraic sum of the head losses around any closed network must be zero. Water flowing in a section of pipe contains three basic forms of energy: energy due to elevation, due to pressure, and due to its motion. Jeppson (1982) presents these three forms of energy in equation form as: E = 92 + - + - [3] where E = average kinetic energy per unit mass in Newton-meter per kilogram, g = acceleration due to gravity in meters per squared second, z = elevation above a reference plane in meter, P = pressure in the pipe in Newton per squared meter, and p = density of the fluid in kilograms per cubic meter . Equation [3] is also known as the mechanical energy equation. When this equation is applied between two points i and j in a pipe section, a mechanical energy balance can be formed. This is shown in Figure 2 and in equation form g 21 + 2 + Bi + W = g zj + E + Ej + EL [4] p where W = work input to the network in Newton per meter per kilogram, and E = energy loss per unit mass due to friction and minor losses in Newton per meter per kilogram. For turbulent flow, the velocity profile is negligible and the kinetic energy is simply E = - [5] Writing the mechanical energy balance equation, [4]. in terms of energy per unit weight and incorporating equation, [5], we find 11 P. N p... N L.» i -—- -- Figure 2. Pipe section (Bralts and Segerlind, 1985) 12 21 + H1 + - + hm = zj + Hj + - + bf [61 where H — is the pressure head; 7 is the specific weight, hm = W/g is the head gain to the network due to external mechanical energy supply, and hf = EL/g is the total head loss due to pipe friction and/or network components. Equations [1] through [6] form the basis for the analysis of water flow through sprinkler irrigation pipes. 2. Hydraulics of Sprinkler Flow The flow of water through an irrigation sprinkler follows the same behavior as flow through any orifice. At the nozzle of the sprinkler the pressure head of the water is converted to a velocity head. The water flows out of the nozzle in the form of a jet and breaks down into drops of water which wet the area within a certain diameter. Pair (1975) and Finkel (1982) have shown that the discharge of a sprinkler depends on the combination of the inlet pressure head and the nozzle cross-sectional 13 area. In equation form Q = CqA\l29h [7] discharge in Liters per second, where Q Cq = discharge coefficient, describing the ratio of actual to theoretical discharge capacities, A = total cross—sectional area of sprinkler nozzle in squared meters, h = inlet pressure head in meters, and g = acceleration due to gravity in meters per squared second. The discharge coefficient for small nozzles commonly varies between 0.95 and 0.98. Some large diameter nozzles have discharge coefficients as low as 0.80. Normally the larger the nozzle, the lower the coefficient (Schwab et al., 1981). 3. Lateral Line Hydraulics To characterize the energy losses occurring in sprinkler irrigation laterals, the Darcy-Weisbach and the Hazen-Williams models will be presented, (Keller and Karmeli, 1975: Pair, 1975; Finkel, 1982; Jeppson, 1982). 14 The Darcy-Weisbach equation is hf = f - - [8] where hf head loss due to friction in meters, f = friction factor, L = length of pipe in meters, < ll velocity on pipe in meters per second D = diameter of pipe in meters, and g = acceleration due to gravity in meters per square seconds. The second equation used to evaluate the friction head loss is the Hazen-Williams equation, which takes the following form 01.852 hf = 10.7 L [9] C1.852 D4.871 where Q pipe flow rate in cubic meters per second, 0 II pipe roughness coefficient, and all other variables are as previously defined. Jeppson (1982) found it useful, when analyzing the flow distribution in large pipe networks, to express the frictional head losses by the exponential formulation 15 hf = K Qm [10] where hf = head loss per unit length of pipe in meters, K = pipe coefficient, Q = discharge of pipe in cubic meters per second, m = empirical head loss exponent which has a value of 2 when the Darcy-Weisbach equation is used and 1.852 when the Hazen-Williams equation is used. The pipe coefficient, K, in equation [10] is a function of pipe roughness, Reynolds Number, length of pipe, and diameter of pipe. If the Darcy-Weisbach equation is substituted in equation [10], then the pipe coefficient K becomes 8f L K = [11] g n2 D5 If the Hazen-Williams equation is substituted into equation [10], then the pipe coefficient K becomes 10.7 x = L [12] C1.852 D4.871 The flow of water in a sprinkler lateral with a number of equally spaced sprinkler outlets will have less friction loss for a given diameter and length of pipe than if the flow was constant for the entire length (Wu and Gitlin, 1975: Pair, 1975). To accurately compute the 16 friction loss in multiple outlet laterals it is necesSary to start at the last outlet on the line and work back to the supply line computing the friction loss between each outlet. Christiansen (1942) simplified this tedious process by introducing an adjustment factor F to correct the friction loss assuming that all the water is carried to the end of the line and that the first sprinkler is one sprinkler riser spacing from the beginning of the lateral. Employing the Hazen-Williams model for friction losses, equation [9]. Q1.852 hf = F 10.7 L [13] C1.852 D4.871 The F value for a finite number of outlets (sprinklers) can be determined by, N l 852 where N = number of outlets (sprinklers) on the lateral line and n = number from 1 to N Jensen and Fratini (1957) modified the above expression for F values to account for the first 17 sprinkler being located one half the sprinkler spacing from the supply line. In equation form, 2 1 N 1 2N _ 1 (1/2 + N1.852 F = n1.852 ) [15] l—‘Ml Wu and Gitlin (1983) calculated F values for both cases, equations [14] and [15]. Figure 3 shows the F values against the total number of outlets for both cases. Case I for placing the first sprinkler one head spacing (equation [14]) and case II for placing the first sprinkler one half head spacing (equation [15]). The shape of the curves in Figure 3 shows that the change of F values with respect to the number of outlets is relatively large for a small number of sprinklers (less than 10). When the number of sprinklers is 10 or more the change in F values is very small. It is also showed that the difference between cases I and II is insignificant when the number of sprinklers is larger than 10. It can also be seen in Figure 3 that when the number of sprinklers is from 5 to 50 for both cases, the F values ranges from 0.36 to 0.40. Since this is practically the range of number of sprinklers for most designs, an average F value of 0.38 can be used in equation [9] for calculating total friction drop without causing any significant error. 18 C1754 F—FACTOR 0.00j’Lfi T I I T 1 T I j r 1 T T 1' T #fi 1 1 T 1 T 1 7 1 NUMBER OF SPRINKLERS, N Figure 3. Relationship between F-factor and number of sprinklers on a lateral line for cases I and II (Wu and Gitlin, 1983). 19 Equation [9] is used to calculate only the total friction drop at the end of the lateral line or submain unit. However, in order to determine the friction drop along the line Myers and Bucks (1972) and Wu and Gitlin (1973) developed an approach to solve for the decreasing flow rate in drip irrigation lateral lines with respect to the length of the line. Bralts (1983) presented a complete development of this approach for the design of drip irrigation submain units. Parts of his work will be presented here, but will be applied to the design of sprinkler irrigation lateral lines. As the total discharge decreases with respect to the length of the line, the energy gradient will not be a straight line but an exponential curve. Assuming an infinite number of outlets and uniform flow the shape of the energy gradient line can be expressed by a dimensionless gradient pressure as derived by Wu and Gitlin (1975) R1 = -—— = 1 - (1 - i)m*1 [16] AH where R1 = pressure drop ratio in meters per meters, AHi = pressure drop in meters at length ratio i, AH = total pressure due to friction in meters, i = length ratio, (L/E) in meters per meters, L = total length of the line in meters, 20 2 = given length measured from the head end of the line in meters, and m = exponent of the flow rate. Figure 4, defines the variables of equation [16]. When the Hazen-Williams equation is used to compute the friction loss, the dimensionless energy gradient line equation [16] can be expressed as: R, = 1 - (1 - i)2°852 [17] The usefulness of the dimensionless energy gradient line resides in its ability to determine the head loss at any point along the lateral line with respect to the original total energy head. The total pressure drop for a sprinkler irrigation lateral line can be determined by using the total discharge as shown by Wu and Gitlin (1975). -an m+l AH L [18] A commonly used form of equation [18] is the modified form of the Hazen-Williams equation, equation [9], where the exponent of the flow rate, m, is equal to 1.852 and the roughness coefficient, C, is equal to 150 for smooth pipes. The resulting equation is 21 r- L j o . 4 ti: H "" I ' | LATERAL LINE LENGTH o 1 A “J 5:: \ AHi AH 35 ESSURE GRADuEtTT'ETJé—JE— - — '— o DISTANCE Figure 4. Lateral line notation and dimensionless gradient line variables (Bralts, 1983). 22 -4 Q1.852 H = 3.50x10 D4°871 L [19] where all the variables are as previously defined. Wu and Gitlin (1983) and Bralts (1983) showed that the pressure variation along a sprinkler irrigation line not only contains the head loss due to friction but also must include the head loss or gain due to slope. The total energy at any section of a lateral line can be expressed from the energy equation as v2 H = z + h + [20] 29 The loss or gain in pressure is linearly proportional to the slope and length of the line and can be shown as follows v2 d( - ) an dz dh 2g ——— = -—- + --- + ----- [21] dL dL dL dL dH where -- = slope of the energy line (-Sf), dL dz -- = slope of the lateral line (-So). dL dh -- = pressure distribution with respect to dL length, and 23 v2 d( - ) 29 -——- = change of velocity head with respect to dL length. The change of velocity head with respect to the length along the a lateral line can be considered small and can be neglected therefore the energy equation can be reduced to = --- + --- [221 rearranging terms, the pressure distribution along a lateral line was shown by Wu and Gitlin (1974) to be dh -—- = $0 - Sf for downslope conditions and dL dh --— = -So - Sf for upslope conditions dL Therefore, the pressure distribution along a lateral line is a linear combination of the line slope and the energy slope. Wu, et al. (1979) expressed the total pressure variation as a function of the original pressure and to the variation due to energy slope and line slope. In equation form hi = Ho — RiAH + Ri'AH' [23] 24 where hi = head for a given length ratio in meters Ho = original pressure head in meters RiAH = head loss due to friction in meters, and Ri'AH' = head loss or gain due to slope in meters Figures 5 and 6 show the pressure variation along a lateral line for downslope and upslope conditions respectively. I As shown in equation [7], the sprinkler flow is directly proportional to the pressure head therefore the sprinkler flow rate at any point along the lateral line will be equal to a combination of equations [7] and [23] or q1 = K(hi)0'5 = K(Ho - RiAH + Ri'AH')0'5 [24] If equation [24] is divided by the sprinkler flow equation for the first sprinkler qo, (qo = K h0.5) then the resulting equation becomes independent of the coefficient K and takes the following form q: = <10 (1 - Riéfl + RILAJI' >°-5 [25] Ho Ho Equation [25] serves to determine the sprinkler flow at any point along the lateral line once the sprinkler flow rate at the original pressure is known. 25 PRESSURE LOSS BY FRICTION FINAL PRESSURE DISTRIBUTION PRESSURE GAIN BY SLOPE 1.0 c: < LU I “J m 8 I— U) LU m a P l .I 14 I I I I I I I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 LENGTH RATIO, I/L Figure 5. Pressure distribution for downslope conditions (Wu et al., 1979). PRESSURE HEAD 26 [- PRESSURE LOSS BY FRICTION FINAL PRESSURE DISTRIBUTION _ . _ . _. PRESSURE LOSS BY SLOPE ._ _______ Ho j ______ _ ‘\\\ -- arr \ . I— \ \ o \.\ . \0 ~ I I I I I I I I I I 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 LENGTH RATIO, l/L Figure 6. Pressure distribution for upslope conditions (Wu et al., 1979). 27 4. Design Criteria The purpose of the sprinkler system is to uniformly distribute the water to the soil surface of the irrigated area. The uniformity coefficient, proposed by Christiansen (1942) is, in general, considered as one of the best design criteria available for sprinkler irrigation design. Christiansen defined sprinkler irrigation uniformity by the equation by 0,: = 100 ( 1 - —) [26] y uniformity coefficient in a percentage basis, where UC Ay = absolute deviation from the mean of the individual observations, and y = mean depth of observation. Most recently, the uniformity of sprinkler irrigation systems has been evaluated employing the statistical uniformity coefficient presented by Wilcox and Swailes (1947). The advantage of this formulation is the use of common statistical parameters such as the coefficient of variation (Vy), which is basically the standard deviation over the mean. In equation form Us = 100 (l ‘ VY) [27] 28 where Us statistical uniformity coefficient and VY = coefficient of variation for the depth of irrigation water, y. Edwards and Spencer (1972), Pair (1975), and Wu and Gitlin (1983) reported that in order to achieve high uniformity system performance it was essential that the pressure variation along a lateral line be held to 20 % so the discharge variation from all the sprinklers along the lateral line would be maintained to be equal or less than 10 %. Wu and Gitlin (1983) expressed the pressure variation along a lateral line as Hmax ‘ Hmin Hvar = 100 [28] Hmax where Hvar = pressure variation expressed as a percentage, Hmax = maximum pressure expressed in meters of head, and “min = minimum pressure expressed in meters of head The pressure variation Hvarr can be determined from the maximum and minimum pressures which can be obtained from the pressure profile using equation [23]. In summary, this section has presented the equations that govern the pressure losses due to friction of the 29 pipes and slope of the ground in sprinkler irrigation laterals. The dimensionless gradient line concept allows for the determination of the pressure or flow rate at any point along the lateral line with respect to the original total pressure or flow rate. B. Methods for the Analysis of Pipe Networks Numerical methods have been Employed in the solution of a wide range of engineering problems. The combined efforts of numerical methods and microcomputers have enhanced the possibility of solving complicated problems which require iterative procedures. Hydraulic design of sprinkler irrigation systems include the solution of nonlinear algebraic equations, which by their nature, must be solved in an iterative manner. Several algorithms have been proposed for solving the nonlinear equations. These include the Hardy Cross method (Cross, 1936), the Newton-Raphson method, first used in pipe flow analysis by Martin and Peters (1963), and the Linear Theory method (Wood and Charles, 1972). More recently, the finite element method has been proposed by Bralts and Segerlind (1985) and Haghighi et al. (1987) as an appropriate technique for the analysis of networks of irrigation pipes. Each of these methods will be reviewed in this section. 30 1. Hardy Cross Method The Hardy Cross method (Cross, 1936) was one of the first methods developed for the solution of pipe networks. This method continues to be very popular and is used today by many practicing engineers. The Hardy Cross method is a simplified version of a method of successive approximations applied to a set of linearized equations (Wiggert, 1986). The method employs a flow corrective technique which uses assumed flow rates, based on continuity, to solve the energy based loop equations of the network system. Applying the continuity principle for each junction, equation [1], becomes IQ = ZQin = zoout = 0 [29] The energy equation in the exponential formulation of equation [10] in terms of the unknown flow rates becomes w = R Qm [30] where W = friction loss for any pipe, including minor losses as equivalent length, R = Pipe coefficient function of pipe roughness, Reynolds Number, length of pipe, diameter, and all the other variables are as previously defined. 31 The resulting energy equations are nonlinear and to obtain a solution require the implementation of a linearization process, employing a Taylor series expansion. An example of a Taylor series expansion is presented in Appendix A. After this process the equations become a1 (901)" + nRi (901)“‘1 ( oi - Qoi ) = o [31] where Qi assumed estimate of discharge and Qoi estimate of discharge from previous iteration. The flow correction is introduced as AQ A0 = Q1 _ Q01 [32] Substituing equation [32] into equation [31] results in the following correction term ‘ Ri Qoi “‘1 Qoi AQ = [33] nRi Qoi n-l The deviation from zero of the assumed flow rate is used to determine the correction value. When the correction value is applied to the assumed flow rate, a better approximation of the true flow is obtained. The successive iterations are carried out until the corrective flows are within a specified tolerance. 32 A possible criterion for convergence could be Qi ’ Qoi Qi < e [34] where e = tolerance value. A second iterative method suggested by Cross (1936) and later developed by Cornish (1939), called the method of balancing heads or nodal method, was described by Barlow and Markland (1969) and by Chenoweth and Crawford (1974). This method solves the nonlinear equations by adjusting the heads at each node until the specified outflows or continuity is obtained. Equation [29] becomes EQin - ZQout = 0 [35] In terms of the head loss W l/m W l/m Z(-)in - z(—)out = 0 [36] R R The method assumes a set of heads which is successively corrected at each node. The correction term then becomes 33 W l/m £(-—-) AH [37] 1 W 1/m - 1 The major disadvantage of the Hardy Cross method is the slow convergence characteristic and in some cases no convergence at all. This is due to the fact that it neglects the change in flow at each node, caused by iterative head corrections at all connected nodes, except for the node at which the flow is being calculated. Several measures, such as the ones suggested by Barlow and Markland (1969), Dilligham (1967), and McCormick and Bellamy (1968), have been implemented for improving the convergence characteristics of this method. Another drawback of the Hardy Cross method is that a reasonably good assumption of the flow distribution in the network is necessary for fast convergence. Epp and Fowler (1970), and Jeppson (1982) reported that the Hardy Cross method, either loop or node oriented, is best used for relatively small, simple networks. 2. Newton—Raphson Method The Newton-Raphson method is an iterative procedure which starts with an estimate of the solution and repeatedly computes better estimates. The Newton-Raphson 34 method is illustrated for a one dimensional case as shown in Figure 7. The goal of this method is to find a solution to an equation of the form F(x) = 0. The procedure starts with an initial estimate which is not too far from the solution, Xm, then, extrapolates along the tangent to its intersection with the X-axis, and takes that as the next approximation. This is continued until either the successive X-values are sufficiently close, or the value of the function is near zero. In equation form X(n+1) = x(n) ' [38] dx new estimate to the solution, where X(n+1) x(n) = initial estimate to the solution, and = correction term dx The application of the Newton-Raphson method to solve simultaneous nonlinear algebraic equations presented by Jeppson (1982) is similar to the one dimensional case described above with the following considerations. IIx) 35 X I X(n+1) X(n+2) XIII) X Figure 7. The Newton-Raphson method for the one dimensional case (Shamir and Howard, 1968). 36 X(n+1) = XIn) - 0'1 F(x(n)) [39] vector of new estimates of the solution, where X(n+1) X(n) = vector of the initial estimate of the solution, D"1 = inverse of the Jacobian matrix, and F(X(n)) = vector of the function of x. The Jacobian matrix D consists of derivative elements, which are derivatives of that particular functional equation. For the x variable the Jacobian matrix is .1 F1171 fifl . . . . . 3_Fl 8X1 3X2 3X' D = 8_F2 3_F2 . . . . . _3_F2 3x1 3x2 axj ar- ar- . . . . . ar- '53:} 53:3 57:} L- .1 If solving the network with the heads as the unknowns the vector x becomes the vector H such that AH(n+1) = AH(n) ’ 2(n) [40] 37 nodal head vector estimate of the where AH(n+1) solution, AH(n) = nodal head vector initial estimate of the solution, and Z(n) = solution vector of the linear system or D(n) z(n) = Fun) [41] Likewise, if solving the network equations containing the loop equations as unknowns the vector x is replaced by the vector AQ such that AQ(n+1) = AQ(n) ‘ 2(n) [42] vector of new estimate of corrective where AQ(n+1) flow, AQ(n) = vector of initial estimate of corrective flow, and z(n) = solution vector of the linear system or 0(a) z(n) = Fun [431 The Newton-Raphson method converges quadratically. This means that when compared to linear convergence of other methods it requires fewer iterations to obtain a solution given a certain tolerance value (Jeppson, 1982). The Newton-Raphson method was first implemented in 38 the solution of pipe networks by Martin and Peters (1963). Subsequently it was used by Giudice (1965), who added a sensitivity analysis and considered more than one fixed head, and by Pitchai (1966), who treated networks with pumps as boundary conditions. Later, Shamir and Howard (1968), also used the Newton-Raphson method to present the basis of the formulation for other types of elements such as pumps and valves. Epp and Fowler (1970) included a method for reducing the storage requirements of the matrix coefficients. The major difference between the Newton-Raphson and the Hardy Cross methods of solution is that the Newton- Raphson method iterates on a set of equations simultaneously, while the Hardy Cross performs iterations on separate equations, one at a time. Because the Newton-Raphson method adjusts the flow rate in all the loops simultaneously, convergence to a solution is much quicker than that obtained using the Hardy Cross method (Wood, 1972). This convergence characteristic is very important for the analysis of networks involving large numbers of pipes. There are two major disadvantages of the Newton- Raphson method. First, it requires the evaluation, either analytically or numerically, of the first derivative of each flow equation with respect to each corrective flow (Lam and Wolla, 1972) and (Bralts, 1983). Secondly, the convergence of the Newton-Raphson method is 39 highly dependent on the initial estimate of the flow distribution in the network. If the initial estimate is sufficiently inaccurate the Newton-Raphson method can lead to slow convergence or in some cases, no convergence at all (Shamir and Howard, 1968; Wood and Charles, 1972; Jeppson, 1982; Bralts, 1983; Wiggert, 1986). Another minor drawback of the Newton-Raphson method is the fact that large networks of pipes, containing numerous components, can lead to the formulation of large size matrices. This limitation can be overcome by employing a bandwidth reduction or renumbering of network elements and junctions, similar to the procedure proposed by Epp and Fowler (1970) and Grooms (1972). Jeppson and Davis (1976) and Chin et al. (1978) described a method for banding the coefficient, or Jacobian matrix, to increase the efficiency of computation. This procedure is a modification of the loop numbering algorithm proposed by Epp and Fowler (1970). Gay et a1. (1978) also presented a two node reordering procedure for water network analysis. This procedure minimizes the required matrix computations, and a sparse matrix technique requiring only off-diagonal non-zero partial derivative elements of the matrix dramatically reduces storage requirements of the matrix equation. 4O 3. Linear Theory Method The linear theory method of pipe network analysis was first presented by Wood and Charles (1972). This technique transforms the loop equations by approximating the nonlinear energy equations in the following form. n n-l ' th = 2K1 Qi = 2K1 Qio Q1 = 2K1 Qi = 0 [44] where hf = head loss in the pipe, Ki = modified pipe constant, Qi = actual discharge, and Qio = approximate discharge. The loop equations can then be expressed as linear equations that when combined with the continuity equations yield "n" linear simultaneous network equations, which can be readily solved for the discharge in each line. An initial approximation of the flow rates can be obtained by assuming that the modified pipe constant is independent of the flow rate such that Ki = Ki. The solution obtained in this manner is very similar to one which would be obtained assuming laminar flow distribution in the network, such that the head loss is assumed to vary linearly with flow rate and be dependent on line length and diameter (Wood and Charles, 1972). By far the most common application of the linear theory 41 method is with loop equations, rather than nodal equations. There are several distinct features of the linear theory method which make it desirable for network analyses. Most importantly, it generally converges in relatively few iterations. This is in part accomplished by averaging the values of the discharge for the previous two iterations and employing the average to calculate the new estimate of flow and the modified pipe constant. This is expressed as: Qi-l + Qi-Z Qio = 2 [45] where Qi_1 flow rate obtained from previous trial and flow rate obtained from trial previous to Qi-2 Qi-l- Also, the method is notable for its ease in programming and use in optimization analysis. In addition, the linear theory method overcomes the principal disadvantage of the Newton-Raphson method or the Hardy Cross method by not requiring an initial estimate of the flow in the network. Wood and Charles (1972) compared the linear theory method with the Hardy Cross and Newton Raphson methods in terms of their convergence characteristics. They found that the linear theory method converges very quickly and 42 more accurately in fewer iterations than the other two methods. Wood and Rayes (1981) reported the reliability of five different algorithms in terms of their convergence and accuracy: PATH, based on the Hardy Cross method for closed loops; SPATH, a modification of the PATH algorithm which calculates all flow adjustments simultaneously (this algorithm is equivalent to the Newton-Raphson method): LINEAR, based on the linear theory method and the linearization of the energy equations; NODE, based on the Hardy Cross method for the solution of nodal equations; SNODE, based on the linear theory method and the linearization of the nodal equations. They found significant convergence problems for PATH, NODE and SNODE methods. Both, the SPATH and LINEAR provided accurate solutions and consistently rapid convergence within a tolerance value of 0.0005 of the exact solution. One of the drawbacks of the linear theory method is that considerably more computer storage is required than needed by the other methods. If used for loop oriented networks as recommended by Wood and Charles (1972) and Jeppson (1982), it uses more equations to solve the network system than are used by other methods. In comparison, the Jacobian matrix in the Newton-Raphson method requires much less storage, since it is a relatively narrow banded symmetric matrix with as many rows as loops. The linear theory method requires 43 computer storage for NxN (number of pipes) plus 1 coefficient matrix while iteratively solving the linearized system of algebraic equations. 4. Finite Element Method The equations which model the flow of water in irrigation pipes are nonlinear in nature and require solution by numerical methods. Norrie and deVries (1978), Segerlind (1984) and others have shown the applications of the finite element method of solution to many types of engineering problems governed by equations of this nature. The finite element method can be formulated using two types of elements: discrete and continuous. Discrete element formulation is generally employed in the analysis of structures. Continuous element formulation is employed in the solution of heat transfer, fluid mechanics, and soil mechanics problems (Segerlind, 1984). In general the finite element method uses an integral formulation to generate the equations describing a problem and employs continuous piecewise equations to approximate the unknown parameters. The advantages of employing the finite element method include: 1) efficient use of computer storage space since the resulting matrix of coefficients is banded and symmetric, and 2) ease of data preparation following a systematic procedure of 44 incorporating the element contributions to the final system of equations. A number of investigators have applied the finite element method to the analysis of pipe network. Norrie and deVries (1978) presented a procedure for solving networks of pipes under laminar flow conditions. In this case, the friction is a linear function of the flow velocity thus allowing the direct formulation of pipe elements without linearizing the flow equations. Henriksen (1984) developed a finite element approach for the analysis of networks of pipes. He employed element conductivities matrices using the method of weighted residuals. Sections of pipes were the only element used and results were based on varying the number of elements that described each pipe. A finite element model of blood flow in arteries including taper, branches and obstructions was developed by Porenta et a1. (1986). This model involves the formulation of a set of nonlinear equations which are transformed into a system of algebraic equations that when solved it yield values of pressure and flow as a function of time and arterial position. Haghighi et al. (1987) formulated tee and wye components based on the linear theory method, nodal head equations, and the application of the finite element method. Bralts (1983) and Bralts and Segerlind (1985) developed a finite element formulation for the analysis 45 <>f drip irrigation submain units. The method is based on the iterative solution of a set of linearized flow equations. They extended Norrie and deVries formulation to include turbulent flow conditions in drip irrigation hydraulic networks. The Galerkin's method of weighted residual was employed to generate the equations representing the two types of elements used: pipe and emitter elements. The following development of the finite element formulation closely resembles the approach used by Bralts and Segerlind (1985). The objective of employing the finite element method is to arrive at an approximate solution for the one dimensional differential equation d2¢ D +Q=0 [46] dx2 Several procedures are available to describe the approximate solution to the differential equation. Among the most popular are: l) the finite difference method, 2) the variational method, and 3) methods that weight a residual. In this case, the Galerkin's method of weighted residuals has been selected to generate a system of equations by evaluating the integral H (12¢ - W(x)(D -- + Q) dx = 0 [47] dx2 0 46 wwhere d = unknown parameters, W(x) = weighting function, and D, Q = differential equation coefficients. Galerkin's formulation employs the shape functions Ni and Nj to define the weighting function for every node. Using a sequence of nodes r,s,t, and u as shown in Figure 8, the residual integral, equation [47] for node 5 becomes Rs = Rs‘e'l) + RS(e) [48] or X Xt s d2¢ d2¢ [Ns(D + Q)](e'l)dx - [Ns(D — + (2)1“)ch [49] dx2 dx2 xr x5 where Rs‘e-l) and Rs(9) represent the contribution of elements (e-l) and (e) to node S. Carrying out the integration of equation [49] for elements (e-l) and (e) results in d¢ D QL RS(e-l) = ”(D — )(e-1)L + " (’¢r + Os) - — [50] dx =Xs L 2 and d Rs(e) = (D _" )(e)L , dx U IO L" (as - ¢t) - -;- [51] 47 (9-1) (9) (e+1) «*0 ‘7 Figure 8. Sequence of nodes and elements (Bralts and Segerlind, 1985). 48 For the special case of steady state flow through pipes, the first derivative term, D(d¢/dx), and the constant Q are always equal to zero. Therefore , the residual equation for node 5 becomes D D Rs = Rs‘e-l) + Rs(9) = -(-¢r + 05) + -(-¢s + ¢t) = 0 [52] L L For a straight pipe element, as shown in Figure 9, the hydraulic analogy to the residual equation [52] begins by applying the energy equation and the continuity equation. The energy equation, considering Vi = Vj and hm = 0 since there are no external mechanical energy supply such as a pump, reduces to P P 21 + — = zj + - + hf [53] Y Y 01' 21 + Hi = zj + Hj + K Qm [54] where Hi and Hj are the downstream and upstream static pressures respectively and all the other variables already defined. Linearizing the exponential friction head loss term: hf = AH = K Qm [55] results in the following relationship 49 p. ---‘ -- N p. N La. Figure 9. Straight pipe element section (Bralts and Segerlind, 1985). 50 0.: K-l/mAHI/m = CpAH [56] equation [54] can be rearranged into Kl/m Q = I (21 + H1) - (Zj + Hj) ]l/m [57] or Q = Cp (Hi - Hj) + Cp (Zi ’ Zj) [58] (Zi + H1) - (Zj + Hj)|l-m/m where Cp = : Kl/m is the linearized coefficient for the straight pipe element and all the other variables are as previously defined. The contribution to the residual equation [52] considers a sequence of nodes as shown in Figure 8. These nodes are separated by elements (e-l), (e) and (e+1). Assuming flow into a node is negative and flow out from a node is positive, the continuity principle applied to nodes 5 and t results [59] I O - Qs(e-l) + Qs(e) and [60] II 0 - Qt(e) + Qt(e+l) the contribution of element (e) to the residual nodal equation is Qs(e) and Qt(e) or Qs(e) = cp(e> (as - at) + cp(e) (zS - 2t) 51 and Qt(e) = -Cp(e) (HS - Ht) - Cp(e) (zs - zt) in matrix notation F j r W QS(e) Cp ”Cp HS CpAz = I I - [61] L _] L 2 where AZ = Zt - 25 The element matrices, shown in equation [61], are assembled employing a direct stiffness procedure which yields a system of equations having the following standard finite element form (Segerlind, 1984): {3(8)} = [149)] {0(8)} - {I491} [521 is the element stiffness matrix, is ana net for con pla fin. 52 {0(9)} = HS Ht is the vector of unknown pressures, and AZ {FIN} = CD = g "CpAZ "g is the element force vector. In summary, this section has shown four numerical analysis techniques employed in the design of pipe networks. The finite element method presents a direct formulation of the system of equations which is constructed by calculating the element's contribution and placing the values in the proper position within the final system of equations. C. Analysis of Hydraulic Components Sprinkler irrigation systems commonly contain a number of hydraulic components such as valves, elbows and tees that contribute a minor loss to the system. Figure 10 is an illustration of the different components noted above. 53 Pipe Element Pump Sprinkler 4 Gate Valve I H i Elbow Tee I] v I --- d ----------1 --I Figure 10. Sprinkler irrigation components. 54 Hydraulic components alter the flow pattern in the sprinkler irrigation system usually creating turbulence which results in head loss in addition to the normal friction loss occurring in the pipes (Jeppson, 1982). According to Finkel (1982), head loss due to hydraulic components may amount to from 2 to 20% of the total head losses in the sprinkler irrigation system and consequently are not always negligible as is often assumed. Head losses occurring in hydraulic components can be expressed in several different ways. In trickle irrigation systems it has been customary to assign pressure losses across fittings as some unspecified percentage of the total pressure loss occurring in the system. In water distribution systems, the procedure is to compute the component losses in one of the following manners: employing the equivalent pipe concept or as a function of the velocity head. The equivalent pipe concept stipulates that head losses occurring in hydraulic components can be determined by forming an equivalent pipe. This equivalent pipe should have the same head loss for any flow rate as the sum of the frictional loss occurring in the pipe and the minor head loss occurring in the component (Jeppson, 1982). The equivalent pipe is formed by adding a length, L to the actual pipe length such that the frictional head 55 loss in the added length of pipe equals the loss from the component. The calculation of L is slightly different depending upon whether the Darcy-Weisbach or the Hazen- Williams equation is to be used. Pair, (1975), Miller, (1978) and others have shown the expression for the head loss occurring in the components of the network system as a function of the velocity head as v2 he = KC " [63] 29 where he = component head loss in meters, KC = component loss coefficient dimensionless, and V2 - = velocity head in meters. 29 The component loss coefficient, KC, is a term which multiplies the velocity head to give the concentrated head loss at the component. The component head loss coefficient may vary according to the flow conditions but is usually convenient to consider it constant for each component (Wood, 1982). Some values normally used for common components are given in Table I. Numerically, Kc, has the same value for English and SI units. Many researchers have investigated the inclusion of hydraulic components in the analysis of networks of pipes. The usual procedure has been to employ one of the three methods of network system analysis namely, the 56 Table 1. Loss coefficients for common components (Wood, 1980). Component KC Globe valve, fully open 10.0 Angle valve, fully open 5.0 Swing check valve, fully open 2.5 Gate valve, fully open 0.2 Gate valve, 3/4 open 1.0 Gate valve, 1/2 open 5.6 Gate valve, 1/4 open 24.0 Short-radius elbow 0.9 Medium-radius elbow 0.8 Long-radius elbow 0.6 45 Elbow 0.4 Closed return bend 2.2 Tee, through side outlet 1.8 Tee, straight run 0.3 Coupling 0.3 45 Wye, through side outlet 0.8 45 Wye, straight run 0.3 Entrance square 0.5 bell mouth 0.1 re-entrant 1.0 Exit 1.0 57 Hardy Cross, the Newton Raphson, or the linear theory method in conjunction with node equations or loop equations. Pitchai (1966) used the Newton-Raphson method in solving a network which included reservoirs and pumps. In the study, the pumps were treated as reservoirs with fixed heads until the network was balanced using the Newton-Raphson method. From this solution, the flow rate at each node connected with a pump is determined. If the flowrate does not satisfy the pump characteristic curve, then the head of the pump is incremented. With this set of new pressure heads at nodes incident to the pumps, the network is rebalanced by the Newton-Raphson method. This process is repeated until the flow rate at each pump node satisfies, within some tolerance, the pump characteristics. Dilligham (1967) employed pumps and reservoirs using the Hardy Cross method of balancing flows. He incorporated the pumps and reservoirs directly into the analysis by assuming that they were connected to the system at joints. Shamir and Howard (1968) included valves and pumps using the Newton—Raphson method and nodal head equations. They found that the convergence of this method can not be guaranteed when any of the characteristic functions of the elements (valves and pumps) in the network do not have continuous derivatives. This theoretical difficulty 58 was overcome in practice by starting the solution with a good initial estimate. Lam and Wolla (1972) considered booster pumps using a modified Newton-Raphson method and linear graph theory. Instead of evaluating and inverting the Jacobian matrix at each iteration, the modified Newton-Raphson method, based on the residue of the functions, uses an iterative equation to update an approximation to the Jacobian and its inverse at each iteration. They compared the modified Newton-Raphson with the standard Newton-Raphson method and found that, when booster pumps were included, the modified method converges faster for larger systems than the standard method. Lemieux (1972) proposed that booster pumps and other elements could be included in the network analysis by using a modified Newton-Raphson method without altering the form of the Jacobian matrix. He obtained a more efficient algorithm by employing a Gaussian elimination procedure. Kesavan and Chandrashekar (1972) employed a graph- theoretic model to solve networks that included pumps and reservoirs. Their approach involved the formulation of two continuity equations, node and loop continuity. The nonlinearities associated with the components in the network were treated as an integral part of the formulation procedure. The graph-theoretic approach was compared with the Hardy Cross method showing a faster 59 convergence and ease of including components in the formulation of the equations. Chenoweth and Crawford (1974) employed the Hardy Cross method of balancing heads in the analysis of networks which included pumps and reservoirs. This method of solution was based on guessing the elevation of the hydraulic grade line at all the nodes in the system. The elevation of the hydraulic grade line at each node, one at a time, is adjusted up or down sufficiently until continuity is satisfied at that node. Calculations then proceed from node to node until all the nodes in the network are covered. To model the pump performance, points from the head-flow characteristic curve are stored on a two dimensional array were they can be conveniently retrieved in each iteration. Wood and Charles (1973) in the closure paper mentioned that a generalized procedure could be implemented to include pumps, reservoirs, valves and other essential components in the analysis of networks by the linear theory Method. Jeppson and Tavallace (1975) found that implementing the above consideration would slow the convergence of the method and suggested a modified approach for the inclusion of pumps and reservoirs in the analysis which greatly improved the convergence. This involved the addition of a pseudoloop to connect two reservoirs and also the inclusion of a linear equation for each pump in the network. I19 [11]] 9X] 60 Dodge et al. (1978) used the Newton-Raphson method and the nodal formulation of the matrix equation to solve networks that included constant discharge pumps, reservoirs, booster pumps, check valves, pressure regulating valves and sprinklers for fire and agricultural use. The formulation of the continuity and energy equations followed Shamir and Howard (1968) analysis. Since network components were included, a node reordering provision had to be implemented such that it reduced the number of computations required. Also a sparse matrix routine was employed which stored only the non-zero elements. Jeppson and Davis (1976) and Jeppson (1982) presented an efficient method for including pressure regulating valves in the network analysis. The introduction of a pressure regulating valve in the network had no significant effect in increasing the number of iterations needed or increasing the amount of execution time required for a solution. Issacs and Mills (1980) employing a Linear Theory method for solving junction heads concluded that minor fittings, pressure regulating valves, and check valves can be readily included in the manner proposed by Jeppson and Davis (1976). Gofman and Rodeh (1981) described a method for incorporating unknown pipe characteristics such as pressure regulating valves and booster pumps into loop 61 oriented hydraulic network solvers. He found that, when head generators (booster pumps) were introduced, the method either may have no solution or may have many solutions. To obtain a satisfactory solution it required that some conditions be meet, such as that the number of head generators be equal to the number of nodes with fixed pressure. Chandrashekar (1980) extended his early work on graph theoretic concepts to include pressure regulating valves, check valves, and booster pumps in the analysis of networks. He employed the junction (nodal) head analysis in the formulation of the system of equations. He found that if several pressure regulating valves or check valves, or both, are present, the method may not yield a correct solution. Numerical problems of oscillations or slow convergence, or both, may also occur. Wood (1981) presented a comprehensive computer program based on his early application of the linear theory method to the analysis of network systems. The computer model is capable of handling any type of pipe system configuration along with any number of components such as booster pumps, pressure regulating valves, check valves and reservoirs. Ohtmer (1983) formulated different network components such as valves, bows, tees, pump, knee, contraction, expansion and diffusers by taking a very 62 distinct approach. He used the Newton—Raphson iteration procedure to solve the nonlinear system of equations and the graph theory concept to determine the initial flow rate estimate. To model the network components, he employed the analogy between the structural network loaded by moments to the flow in pipe networks. With this procedure, the flow rate is the equivalent variable of the moment as the pressure is to the rotation. This was made possible by applying mesh (force) method to include the components. D. Summary In summary, the literature review has shown that the finite element formulation can be employed in the design and analysis of pipe networks. Furthermore, a need has arisen to include the pressure losses occurring in elbows, tees, valves and also the contribution of pumps in the design process. The proposed research will focus on the formulation of equations that model these components, employing the finite element method. I I I . METHODOLOGY A review of the literature indicates the need for better sprinkler irrigation design so that water and energy may be used in the most efficient manner possibly. In order to achieve maximum efficiency it is necessary to take into consideration the losses occurring in the hydraulic components of sprinkler irrigation systems. The review of the literature also shows the difficulty of incorporating these hydraulic components when employing any of the conventional solution techniques of pipe network analysis. This suggests the need for a better formulation where in a systematiC‘ manner such hydraulic components can be readily incorporated in the analysis of sprinkler irrigation systems. A. Research Approach Based upon the need to obtain a more accurate design of sprinkler irrigation systems including the effects of losses occurring in the hydraulic components, the following procedures are proposed for attaining the research objectives. 63 64 Objective 1. To develop finite element formulation of essential sprinkler irrigation components such as elbows, tees, gate valves, sprinklers and pumps. To achieve objective 1, various sprinkler irrigation hydraulic components will be formulated employing the finite element methodology. A computer model will be designed based on the finite element formulation of the hydraulic components and on the application of the dimensionless energy concept to sprinkler irrigation design. The purpose of the computer model will be to provide reliable values of pressure and discharge in order to satisfy the sprinkler irrigation system design. Objective 2. To demonstrate the potential of applying the finite element formulation to the design of sprinkler irrigation systems by comparing it to a linear theory approach. To attain objective 2, the finite element based computer model will be compared to a commercially available linear theory computer model employed in the solution of pipe networks. Specifically, both models will be compared for reliability and accuracy of solution. 65 B. Theoretical Development The theoretical development section is composed of two parts. In the first part, the finite element formulation of a number of sprinkler irrigation components is presented. In the second part, an example implementation is presented. The following development of the finite element formulation is an extension of the approach outlined in the literature review section. The analysis is based on the mechanical energy balance equation and the contribution of each of the system components to the nodal equation. Applying the mechanical energy balance between two points in a network system, Figure 11, (neglecting the velocity component) is Pi Pj 21 + -—— + Hp = zj + -—- + hf + hc [64] 90 90 where the subscripts i and j denote the downstream and upstream conditions respectively and all other variables are as previously defined. Equation [64] can be rearranged into the different contributions of each element to the system KC Q2 29 A2 (21 + H1) - (zj + Hj) = KQm + Hp [65] 01' Figure 11. Pipe 66 Sprinkler 21 p---------------- F----- 23' section and system components 67 2 KC Q AH = KQm + ————3— - Hp [66] 29A where A is the area of the element and AH is defined as AH = (21 + H1) - (Zj + Hj) Equation [66] can be separated further into three equations: AH = KQm Pipe Section [67] KC Q2 AH = -—-—- System Components [68] 29A2 AH = -Hp System Pump [69] starting with equation [67] it can be rearranged into Q = K‘l/m AHl/m [70] equation [70] can be linearized in the following form AH Q g K-l/m AHl/m ___. AH Q = K-l/m AHl/m+l-l resulting in the following equation 68 Q = Cp AH [71] equation [71] can then be rearranged into Q = Cp (Hi - Hj) + Cp (zi - zj) [72] (21 + H1) - (Zj + Hj) (l-m)/m = 7 where Cp Kl/m [ 3] is the linearized coefficient for the straight pipe element. Equation [68], representing the system component contribution, can be rearranged into )3ch2 2 = RC Q2 = AH [74] 29A or RCl/2 Q = An“2 [75] and Q = cpC (Hi — Hj) + cpC (zi - zj) [76] AHl/Z where Cpc = [77] Rel/2 is the linearized coefficient for the system components. 69 The effects of the pump in equation [69] can be described in three basic forms: First, the effect of the pump can be specified depending on the useful power it puts into the system (in horsepower or kilowatts). The useful power refers to the actual power which is transformed into an increase in pressure head and kinetic energy of the liquid as it passes through the pump. The useful power can be computed from a typical head—discharge data set using the following equation P H = - [78] P OQQ where Hp = operating head in meters, P = useful power in kilowatts, Q = discharge in cubic meters per second, 0 = density in kilograms per cubic meters, and g = acceleration due to gravity in meters per squared second. Rearranging equation [78] results into P AH = - [79] 090 or Q = Cpp (Hi - Hj) + Cpp (Zi - Zj) [80] 70 P [81] where C = - PP AHpg is the linearized pump coefficient. Alternately, a pump can be described by points of operating data. A polynomial or an exponential curve can be fit to this data to obtain a pump characteristic curve describing the pump operation as Hp = H1 — ch [82] where H1 cutoff head in meters, c and m coefficients derived from each particular pump application, and all the other variables as previously defined. Equation [82] can be rearranged into AH = - H1 + CQm [83] or (AH + Hl)l/m = cl/m Q [84] where and — |(AH + Hl)(1-m)/m _ cl/m Cpp [86] is the linearized pump coefficient. 71 A third method of incorporating the effects of a pump into the system of equations is by specifying the pump discharge pressure. For this application the pump discharge is considered at a fixed grade node, where both the pressure and the elevation are known values. The pump contribution can be written as AH = Ho => AH = + 20 [87] where H0 is the fixed grade node. The pump contribution in this case is accounted for in the forcing vector as ( ) _ (21 + H1) - (zj + Hj4(l-m)/m where Cpp = Kl/m * [89] is the linearized pipe element coefficient in which the pump is located. The sprinkler element can be considered as a separate component. Rearranging equation [68] and neglecting the elevation difference across the sprinkler results into ( ) KCQZ [ ] H- - H- = -————— 90 1 3 29A2 72 or and (Hi - Hj)l/2 = Kl/ZQ [92] where Q = cSp (Hi - Hj) [93] and (Hi - Hj)1/2 Csp = [94] Kspl/Z is the linearized coefficient for the sprinkler element. Combining all the linearized coefficients into one equation results into zc (AH) [95] IO u 01' Q = (Cp + CC + Cpp + Csp) AH [96] which gives the linearization of all the elements as shown in Figure 11. The finite element method utilizes the concept of an element stiffness matrix and an element force vector to construct the system of equations. The element matrices for the pipe element is developed below. Considering a sequence of nodes as shown in Figure 12. These nodes are separated by elements (e-l), (e) and 73 (9“) (9) (E+I) U! (*4) C Figure 12. Sequence of nodes and elements (Bralts and Segerlind, 1985). 74 (e+l). Assuming flow into a node is negative and flow out from a node is positive, the continuity principle applied to nodes 5 and t results [97] I O - Qs(e—1) + Qs(e) and [98] II 0 - Qt(e) + Qt(e+l) the contribution of element (e) to the residual nodal equation is Qs(e) and Qt(e) or Qs(e) cp(e) (n5 - at) + cp(e) (zs - z.) [99] and Qt(e) -Cp(e) (Hs - Ht) + Cp(e) (zs - zt) [100] in matrix notation f r- '1 QSIE) Cp 'Cp HS Cp AZ [101] __A II :1: f? I L L 1 where A2 = Zt - ZS Likewise, the components' (elbows, tees and valves) contribution to the residual nodal equation is Qs(e) QtIE) 75 - [102] The pump contribution is also defined in the same format as 1 = - [103] Finally, the sprinkler contribution to the residual nodal equation can be written as Qs(3) QtIE) HS cp Az - [104] A direct stiffness procedure, similar to the one employed in structural analysis, is used to incorporate the element matrices shown in equations [101], [102], [103], and [104] into the final system of equations. To demonstrate the direct stiffness procedure a sprinkler irrigation system as shown in Figure 13 has been selected. The sprinkler system consists of a submain and eight laterals. The submain contains seven 76 G) ----- InpUt Node [><] ----- Gate Valve C) """ Sprinkler E] ----- Elbow - Tee c)15 11 7 q) 3 §?(15) (11) (7) ii (3) A (16) (14) (12) (10) (8) 17 16 14 12 10 8 6 4 2 2E (13’ %§ (9) j; (S) (l) l (>13 (D 9 C) s 1 Figure 13. Example sprinkler irrigation system. 77 tees and one elbow. Each lateral contains one sprinkler and one gate valve. The input has been considered as a fixed grade node where pressure and elevation are known quantities. The direct stiffness method is very simple to apply and only requires the knowledge of the location of the elements defined by two nodes. The contribution of each element in the system of pipes and components is accordingly added or subtracted at the proper location in the stiffness matrix. An example of the solution matrix is shown in Figure 14. The direct stiffness procedure yields a system of equations which has the general matrix form: [K] {H} - {F} = {0} [105] The vector {H} contains the element nodal pressures, the stiffness matrix [K] contains the algebraic summation of each element contribution, and the vector {F} contains the element contribution to the force vector. 7B ”—2 lm~v 000.00.00.00... 6 O O I L__, .n—. .nu. ._:o .n. .mzo .n. 00‘ .¢:6 .m~. ..:o .n:6 .-.o N a-. x :l! a: 1:; a. 2 . 9 CU 0 .9 . .U 0 .o.. .o.. .v.. 0 0 0 .0. .0. .0 .0. .0. .v. o... a... .o.. .u-. .o.- ...- O a. . a . u . v v v .o.. .o.. .o-. .o.. u .0 0 a. .U I .VI 9 . ”U. U . .-.. .~.. .~.. . .9 0 At. .a.. .-.. a 0. . nY .... .~.- 0 0 0. O... .. O. 0 v .0. an. .—- .g. a c 00 o .u o Au :- ... .a. o. o. .- ., ... .... .....,. a .l Example solution matrix. Figure 14. IV. RESULTS AND DISCUSSION The results and discussion section consists of a comparison of the data generated by both finite elements and linear theory models. Data analysis and representation will be performed using the PLOT-IT statistical package. Linear regression analysis will be performed to determine the correlation between the junction pressures and sprinkler flows for the finite elements and linear theory models. Also, the accuracy and reliability of each model will be assessed in this section. A. Computer Models A computer program (DESIGNER) developed by Bralts and Segerlind (1985), which uses the finite element method to assist in the analysis and design of drip irrigation submain units, was modified to incorporate the element matrices derived in the methodology section. The finite element based computer model (SPIRR-FE, see appendix B) was compared with a computer model based on linear theory developed by Wood (1980) (see appendix C). The finite element based computer program (SPIRR-FE) solves the hydraulic network problem by obtaining values 79 80 of pressure and discharge, at the junction and sprinkler nodes, and employing them to satisfy the sprinkler irrigation system design. The SPIRR-FE computer program uses the dimensionless grade line concept as presented in the literature review to generate initial estimates of both junction pressures and elevations. The linear theory model, as presented in the literature review section, utilizes a different approach to approximate the initial estimates of flow. Wood's computer program calculates initial flowrates assuming a modified pipe line constant which is made independent of the flowrate. The solution obtained in this manner is very similar to one which would be obtained assuming laminar flow distribution in the system. The results of this section will show that both techniques for computing the initial flow distribution in the hydraulic system give very reasonable values as it does approximately compute a laminar flow distribution. This gives results which are feasible estimates of the turbulent flow distribution existing in the sprinkler system. B. Model Comparisons Both the finite element and the linear theory computer programs provide empirical solutions. For this reason, comparison of their solution should be made 81 considering a variety of initial conditions. A group of comparisons were conducted which attempted to cover a wide range of possible solid set sprinkler irrigation hydraulic designs. Five different slope conditions were employed in the comparisons. They ranged from two percent downhill to two percent uphill. Also a zero percent slope was considered. Four different values of component KC coefficients were also selected. They varied from zero to three. The purpose of this type of evaluation was to examine the impacts of the system components and the sloping ground conditions in the overall sprinkler irrigation system design. The comparisons were further divided into two general situations. The first evaluation procedure consisted of comparing the results of head and flow generated in a sprinkler irrigation system, considering the input node of the system as a fixed grade node. The second evaluation procedure consisted of comparing the same system, but including a pump as a source of head input into the system. The sprinkler irrigation system selected for the first evaluation is shown in Figure 15. It consists of a submain and eight laterals. The submain line has a diameter of 203.2 millimeters (8 inches) and a total length of 487.7 meters (1600 feet). The submain line also contains six tees and one elbow. The lateral 17 82 C) ----- Input Node {3 ----- Sprinkler C><fl ----- Gate Valve [3 ----- Elbow - Tee (P15 1) 11 (P 7 . ‘5 SLOPE = 0% LIJ I E 7-25- COMPONENT - E COEFF. Kc = 0.0 2 x’ :3 I,’ [1:1] 7500 1 1 1 7.00 7.25 7.50 7.75 FINITE ELEMENT SPRINKLER FLOWS (l/s) Figure 22. Comparison of finite element and linear theory solution for sprinkler flows with 0 percent slope and Kc equal to 0.0. 8.00 94 BIN) 1 1 I I E LINEAR RECRESSION , t: Y = -0.034 + 1.006tX ,I’ 2 ’1 (3) 7.50_ R— 1.000 x, q .J ’I u. a: uI .1 § 55 7L00d - O. U) >- g SLOPE = 1% uphIII E 6.50- COMPONENT - 01 , COEFF. Kc = 1.0 :1 I,” [1:1] BIN) 4r I T 6.00 6.50 7.00 7.50 FINITE ELEMENT SPRINKLER FLOWS (l/s) Figure 23. Comparison of finite element and linear theory solution for sprinkler flows with 1 percent upslope and Kc equal to 1.0. 8.00 9S ”100 T 1 *r n 1 1 T I, 7:? LINEAR RECRESSION / > 950- ” - Z - Y = 0.010 + 1.000.x 2 ’l % 9.00-l R — 1.000 III 4 _J I LI. ’I’ 35 8.50~ /’ 4 ._.l E 55 ELOO- - CL 0') E 7.50~ I. - 8 /’ SLOPE = 2% downhill E 7-00- COMPONENT - a: COEFF. Kc = 2.0 :25 6.5O- .. :3 I,” [1:1] 6.00 T i T T T r f 6.00 6.50 7.00 7.50 6.00 8.50 9.00 9.50 10.00 FINITE ELEMENT SPRINKLER FLOWS (l/s) Figure 24. Comparison of finite element and linear theory solution for sprinkler flows with 2 percent downslope and Kc equal to 2.0. 96 81“) I 1 T I *1 1 1 ’, 3 LINEAR REGRESSION 5 7-50‘ Y = 0.010 + 1.000:x d ‘g 700- R’= 1.000 ’ _ E‘.’ 33 6.504 ‘ .J E E 6.00- T 8 E 5.50- T 8 / SLOPE = 2% uphill E 5.00-1 COMPONENT - E COEFF. Kc = 3.0 2 4.50" ,” :3 I,’ [1:1] 4.00 Figure 25. T T T T T’ T— T 4.00 4.50 5.00 5.50 6.00 6.50 7.00 7.50 8.00 FINITE ELEMENT SPRINKLER FLOWS (l/s) Comparison of finite element and linear theory solution for sprinkler flows with 2 percent upslope and Kc equal to 3.0. 97 Table 4. Comparison of junction heads between the finite element and linear theory models SLOPE Kc AVERAGE PERCENT (%) DIFFERENCE 2 0 0 0.011 1 0 0 0.004 0 0 0 0.005 -1 0 0 0.006 -2 0 0 0.008 210 """"""""" 6:666 """ l l 0 0.002 0 1 0 0.012 2 l 0 0.004 -2 1 0 0.007 2200035 """" 1 2.0 0.035 0 2 0 0.041 -1 2.0 0.042 -2 2.0 0.047 2300078 """ 1 3 0 0.084 0 3 0 0.087 -1 3 0 0.095 98 maximum percent difference or error for the junction heads in the order of 0.1 percent. This is probably due to the accuracy of the initial estimates and to the convergence criteria employed by each model. Also, this error can be partially attributed to the different methods employed by each model to solve the final system of equations. The results of the total sprinkler flow comparisons between both models is presented in Table 5. This table shows the percent difference varying from a low of 0.031 percent (for a sprinkler system with a 2 percent down hill slope and with component coefficients equal to 0.0) to a high of 0.194 percent (for a sprinkler system with 2 percent down hill slope and a component coefficient equal to 3.0). These results suggest a minimum degree of variation between the sprinkler discharges in each model. The results of Table 5 also show the components contribution to the sprinkler system design. As the component coefficient Kc is increased, the total system flow rate decreases. Both models simulate this condition with a high degree of accuracy. Figures 26 and 27 show the hydraulic grade line for two of the cases studied; up hill and down hill respectively. These figures demonstrate the effects on the hydraulic gradient line due to the main line slope, the pressure on the laterals and the system components. It is apparent from Figure 26 that the pressure loss 99 Table 5. Comparison of sprinkler flow rate between the finite element and linear theory models SYSTEM FLOW RATE (liters/second) SLOPE Kc PINITE ELEMENT LINEAR THEORY PERCENT (%) MODEL MODEL DIFF. 2 0.0 51.69 51.71 0.039 1 0.0 55.53 55.55 0.036 0 0 0 59.05 59.07 0.034 -1 0.0 62.34 62.36 0.032 -2 0.0 65.43 65.45 0.031 "5"""6'6m'""66??1""""""66T66 """" 6'66? 1 1.0 54.47 54.50 0.055 0 1.0 57.92 57.95 0.052 -1 1 0 61.13 61.17 0.065 -2 1.0 64.16 64.19 0.047 "6""“666 """" 66:66 """""" 66T66""""6TI66' 1 2 0 53.45 53.51 0.112 0 2.0 56.83 56.89 0.106 -1 2.0 59.98 60.04 0.100 -2 2.0 62.94 63.01 0.111 "6"”"666 """" 66:66 ”””””” 66T66""""66666' 1 3 0 52.48 52.57 0.171 0 3 0 55.79 55.89 0.179 -1 3.0 58.87 58.98 0.187 100 400 1' j j 1 1 fi 1 1 fl 1 j ‘1' 1 1 1 1i 1 j 1 1 j 1 1 1 J - HYDRAULIC GRADIENT LINE 4 320- A ’o‘ ‘ ‘ 0. .1 . 5 - . o 2401 PRESSURE REQUIRED FOR LATERALS 1 f5 - 1 I: 1 . 1&1 . . D 160- - a .1 MAIN LINE SLOPE 1 LIJ :1 II E ‘ SLOPE = 2% uphill 1 80‘. COEFF. Kc = 3.0 i O 1T 1 i 1r T 1 1' 1 T T 1r 1" 1 1 T111 1 1 T j i 1 I 0 100 200 300 400 500 MAIN LINE LENGTH (meters) Figure 26. Hydraulic grade line representation for 2 percent uphill and Kc equal to 3.0. 400 101 L l IA (:1 N ‘1" PRESSURE HEAO (KPO) j 1 1 1 I 1 j\ COEFF. KC = 1fi111fifi—1Tj1 A HYDRAULIC CRAOIENT LINE llkL ll 1 L l 1 L L14 14 A v A PRESSURE REQUIRED FOR LATERALS SLOPE = 2% downhill 21) MAIN LINE SLOPE \\. l l l O +1.1 0 Figure 27. fi‘rfij 1 j j—T'T 1" j Tfi 100 200 300 MAIN LINE LENGTH (meters) fifiTf 400 500 Hydraulic grade line representation for 2 percent slope downhill and Kc equal to 2.0. 102 due to the combination of the friction in the lines and the system components is less than the gain in pressure due to the slope of the main line. This result suggest a hydraulic gradient line with a gradual slope along the main line length. From figure 27, it can be seen that the combination of uphill 510pe and pressure loss due to friction reach an equlibrium point at approximately 180 meters. The hydraulic gradient line in this case is more horizontal than in the previous Figure suggesting larger junction head pressures towards the end of the sprinkler system. The second evaluation procedure was a comparison of both models using the same system layout but with the addition of a pump in line 1 between nodes 16 and 17. Figure 28 shows the sprinkler irrigation system layout and the components. The pump selected in this part of the evaluation procedure was capable of providing a pressure head equivalent to 119.6 Kilopascals (40 feet), in addition to the existing conditions of head and elevation in the system. Tables 6 and 7 show the results of the correlation and linear regression analysis of the junction heads and sprinkler flows respectively, with a pump in the system. Figures 29 through 32 show this relationships in a graphical form for the four cases selected. From these Tables and Figures it is demonstrated the high degree of .103 I>< ----- GOLD V0121. C) ----- Input Node Q3 ----- Sprinkler E] ----- Elbow -'hy: (>15 7 I }'(15) I (11) V7 (7) I? (3) .3 l1 /\ /\ (10) (8) (6) (4) (2) A——‘EI—”—B-—*"“E}—' {SI—“"E—T 4 2 (16) (14) (12) o l—l r—1 L_J LJ 17 1o 14 12 10 a 5 I7 ‘7 )7 (I I I I .- /1 (13’ /> (9) 1 (1) <3 13 (D 9 £3 5 Sprinkler irrigation system layout with Figure 28. FNNWP- 104 Table 6. Results of the correlation and linear regression analysis of the junction heads with a pump in the system. Slope Component N Regression Equations R2 (%) Coeff. KC ”'E”"""3T6 """" é """ §’;'"6T§§§I6f§§5¥§""’1T666 l 1.0 8 Y = —0.792+1.003*x 1.000 0 0.0 8 Y = -0.991+1.003*x 1.000 -2 2.0 8 Y = -1.739+1.005*x 1.000 Table 7. Results of the correlation and linear regression analysis of the sprinkler flows with a pump in the system Slope Component N Regression Equations R2 (%) Coeff. KC ""5""""3T6 """" 5 """ Q’L”6T6£§IST3331§""’1T666 l 1.0 8 Y = -0.047+1.006*X 1.000 O 0.0 8 Y = -0.015+1.002*x 1.000 -2 2.0 8 Y = -0.097+1.011*x 1.000 105 LINEAR THEORY JUNCTION PRESSURES (KPo) Y = -O.792 + 1.0031-X R’= 1.000 4 I,” [1:1] 1 j l 1 I f I V W W 1 I U U 1 j fir fi— LINEAR REGRESSION SLOPE = 1% Uphill COMPONENT COEFF. Kc = 1.0 PUMP HEAD= 119.6 KPo: L I l l L l l l l l I l l J l I l Figure 29. j “I I j I j j j 1 j —r j fit 2% ' 290 31 O 330 350 FINITE ELEMENT JUNCTION PRESSURES (KPO) Comparison of finite element and linear theory solution for junction heads with 1 percent upslope and Kc equal to 1.0 with a pump in the system 106 E 410 d T Ifi j ‘l U T T I Iii T 1 jfi fl ‘I j 1 ‘l j jiT 1 Y I ’1" 5 ; LINEAR REGRESSION j a 400-: Y2: —1.739 + 1.005fix _': g ‘ R = 1.000 I U) ‘ -I m .1 3: 39°: 1. ‘L - 1 2‘3 3 1 z: ‘ j 3 -I '3 4 1 E 370.1 SLOPE = 2% downhill -; 8 f COMPONENT j E 350-? COEFF. KC = 2.0 -% E : [1 1] PUMP HEAD: 119.6 KPo; :—z—’ 350 ’:jfiT T. Tfifi T 1 1 1 1 I f 1 W W T 1fifiFTW 1 T I d 350 360 370 380 390 400 410 FINITE ELEMENT JUNCTION PRESSURES (KPO) Figure 30. Comparison of finite element and linear theory solution for junction heads with 2 percent downslope and Kc equal to 2.0 with a pump in the system. 107 10-00 T T n 1 1 9’ E LINEAR RECRESSION m 9050‘ Ya= -0.047 + 1.006:x _ E R = 1.000 d I m 9.009 - Lu _1 :5 E 6.50J - (L U) E 8-00-1 SLOPE = 1% Uphill - .25 COMPONENT g 7.504 COEFF. Kc -= 1.0 - g PUMP HEAD= 119.6 KPO —-' ’1’ {1:1} 7-00 T f T —I T 7.00 7.50 8.00 8.50 9.00 9.50 10.00 FINITE ELEMENT SPRINKLER FLOWS (I/s) Figure 31. Comparison of finite element and linear theory solution for sprinkler flows with 1 percent upslope and Kc equal to 1.0 with a pump in the system 108 LINEAR RECRESSION Y = -0.097 + 1.011.x R’= 1.000 SLOPE = 2% downhill COMPONENT j COEFF. Kc = 2.0 PUMP HEAD= 119.6 KPO I I ’l” [111] 1CL00 ’6? > ‘é’ o 9.75- 11‘ a: u: .1 32‘ E 9.50H a. U) )- a: C) u: E; ELZSH a: 5 Z .1 £100 91) Figure 32. 9.25 9.50 935 FINITE ELEMENT SPRINKLER FLOWS (VS) 0 Comparison of finite element and linear theory solution for sprinkler flows with 2 percent downslope and Kc equal to 2.0 with a pump in the system 1CLOO 109 correlation between the finite element and linear theory solutions. Table 8 shows the comparison between the junction heads for each model with a pump in the system. The results of this table suggest that, for the four cases selected, the percent difference or error did not exceed 0.022 percent. This corroborates the assumption that a solution can be considered to compare favorably if the percent difference between the junction heads did not exceed 0.1 percent. Table 9 shows the comparison between the total sprinkler flow for each model with a pump in the system. The highest percent difference or error between the models was found to be equal to 0.142 for a system with a 2 percent up hill slope and with component coefficients equal to 3.0. This value seems to be in line with the assumption that a reasonable solution can be obtained provided the sprinkler flows did not varied by more than 0.2 percent. Figures 33 and 34 show the hydraulic grade line for the up hill and down hill cases with a pump in the system. These figures demonstrate the effects of the pressure added by the pump to the system. The impacts of the overall increase of pressure in the system is reflected in the higher level attained by the hydraulic grade line. 110 Table 8. Comparison of junction heads between the finite element and linear theory models with a pump in the system SLOPE Kc AVERAGE PERCENT (%) DIFFERENCE 2 3.0 0.022 1 1.0 0.002 o 0.0 0.003 -2 2.0 0.019 Table 9. Comparison of sprinkler flow rate between the finite element and linear theory models with a pump in the system SYSTEM FLow RATE (liters/second) SLOPE KC FINITE ELEMENT LINEAR THEORY PERCENT (%) MODEL MODEL DIFF. 2 3.0 63.16 63.25 0.142 1 1.0 68.52 68.56 0.058 0 0.0 72.72 72.75 0.041 -2 2.0 75.07 75.14 0.093 111 ii 1 1 Ifij Ti lj j Tfi 1 T—Vj 1 1 T 1—‘ T 500- .11 A V A A HYDRAULIC GRADIENT LINE 1 I II 1 l l 4'11 4; O O - v F ‘ ‘ v f v U 0 O N O O LlLLlllAllllAlLlllllAll /RESSURE REQUIRED FOR LATERALS / SLOPE = 2% downhill COEFF. Kc = 2.0 PRESSURE HEAD (KPO) 100 MAIN LINE SLOPE LLIIJ+IJJ1LILLH PUMP HEAD= 119.6 KPoj TTT‘TTjjf‘TfifiTiTjjjfir'Tfirfir‘TT O 100 200 300 400 500 MAIN LINE LENGTH (meters) 0 Figure 33. Hydraulic grade line representation for 2 percent downslope with Kc equal to 2.0 with a pump in the system. 112 W V W T 500 N (:1 4F O O O O O O l [LL11 lAl l LlLl lJ l l 1 .Al 1 I l L PRESSURE HEAD (KPc) a... O O 0 j v HYDRAULIC GRADIENT LINE IWWI 1W WW W W 1 W 1 T—IW' W V I A v A A A v f ! PRESSURE REQUIRED FOR LATERALS MAIN LINE SLOPE SLOPE = 27. uphill COEFF. Kc =- 3.0 PUMP HEAD= 119.6 KPo JIIIIIALILIIIILILLJIL llll WWWW 0 TWWW WWWW‘WWWW 100 200 500 MAIN LINE LENGTH (meters) I I 1 1 T 400 Figure 34. Hydraulic grade line representation for 2 percent upslope with Kc equal to 3.0 with a pump in the system. 500 113 B. Accuracy and Reliability The reliability of both computer models was examined by evaluating a large number of sprinkler irrigation designs employing each algorithm and assessing the consistency of each model in arriving to a solution. Under the 24 simulations studied, an acceptable solution was always attained. The accuracy of a solution is a measure of the exactness to which the basic hydraulic equations are satisfied for a given irrigation design. The accuracy of the computer model solutions was determined by comparing the sprinkler flows and the junction head. The solutions were considered to compare favorably if the percent difference between sprinkler flow rates did not exceed 0.2 percent and the percent difference between junction heads did not exceed 0.1 percent. This criteria was susscessfully meet in all the simulations studied as shown in Tables 5 and 9 for the sprinkler flows and in Tables 4 and 8 for the junction heads. The convergence criteria is a measure of the speed at which a relatively stable solution can be obtained. In this particular case, the convergence criteria employed by each computer model was different. The linear theory computer model employed the change in flowrate between successive trials to Check for CIJnvergence. The specific convergence criteria employed 114 in this method is referred as the relative accuracy. This criteria roughly states that when the average change in flow rates between successive trials is less than 0.5 percent, the calculation cease. Although this criteria is more stringent than others normally applied in practice, it does not however, assure that the flow rates are within 0.5 percent of the correct values. The finite element method arrives at a solution with a relative accuracy of 0.03 Kilopascals (0.01 feet). The model uses the criteria that the sum of the total pressure variations in all the nodes has Changed less than a predetermined relative accuracy, in this case 0.03 Kilopascals. C. Summary Discussion The results and discussion section have demonstrated the capabilities of the finite element computer model to simulate a wide variety of solid set sprinkler irrigation systems. The results comparing the finite element computer model (SPIRR-FE) and the linear theory model were very positive. The results strongly support the use of the dimensionless energy gradient line concept for initial estimating the junction pressures and elevations. The reliability of the algorithm employed in the irrigation system solution is of great importance. Failure to obtain a solution is an inconvenience and the 115 failure to recognize a poor solution may be even a greater problem because this may lead to poor design or management of the irrigation system. No limitations to the SPIRR-FE computer program were found for the range of analysis reported. The results of the above comparisons clearly show the validity of the finite element solution. V . CONCLUSIONS AN D RECOMEN DAT I ON S The objectives of the proposed research have been addressed in full. A computer program which uses the finite element method to model essential sprinkler irrigation system components was developed. The computer program was compared to a linear theory model which demonstrated the applicability of the finite element formulation to the design of sprinkler irrigation systems. The computer based finite element solution results in a symmetric banded matrix which greatly reduces the computer storage. The specific conclusions of the research are: l. The finite element method can be used to model sprinkler irrigation system components such as elbows, tees, gate valves, sprinklers and pumps. 2. The finite element based computer model, when compared to a linear theory model, showed very good correlation for junction pressures and sprinkler flows. 116 117 3. The finite element method converged to reliable solutions provided reasonable initial pressure estimates were obtained by employing the dimensionless gradient line concept. Recommendations for further research include: 1. Extend the finite element methodology to cover hydraulic network analysis including systems with loops. 2. Apply optimization techniques to the finite element model to obtain the best design possible given a set of initial conditions. APPENDI CES APPENDIX A TAYLOR SERIES EXPANSION TAYLOR SERIES EXPANSION In the Review of Literature section all the methods presented in the analysis of pipe networks employed some form of linearizing the nonlinearities associated with the energy equations. One of the most widely used linearizing technique employed in these methods is the Taylor series expansion. The mathematical basis of this technique suggests that if a power series £x(n) (x-x(n)) converges to a function f(x) then the coefficients of the power series should be determined by the values of that function and its successive derivatives. Applying this principle to the Newton-Raphson method suggests that in order to arrive to a solution of f(x) = 0, it is necessary to determine the tangent of the curve at the first approximation (x(n)). Mathematically the evaluation of f(x) = 0 can be written as f"(c) f(x) = f(x(n)) + (x-x(n))f'(x(n) + (x-x(n))2 [A.l] where C(n) lies between x(n) and x. Solving for x results in f(X(n)) 2 f"(c) x = x(n) - -——-- - (x-x(n)) ---—- [A.2] f'lme 2f'(x(n)) 118 119 01‘ f"(r.) 2f. (X(n)) [11.3] x = x(n+1) ’ ("”‘(n))2 In this case the error of the (n+1) estimate is proportional to the square of the previous error. APPENDIX B SPIRR-FE PROGRAM CODE {******************************************************} PROGRAM SPIRR-FE(INPUT,LSST); I SPIRRE-FE THIS PROGRAM CALCULATES THE JUNCTION HEADS AND SPRINKLER FLOW RATES OF A SPRINKLER IRRIGATION SYSTEM EMPLOYING THE FINITE ELEMENT FORMULATION. THE COMPUTER PROGRAM WAS WRITTEN IN TURBO-PASCAL. BY LUIS ALFONSO SALDIVIA MICHIGAN STATE UNIVERSITY AGRICULTURAL ENGINEERING DEPARTMENT 1988 {******************************************************} VARIABLES USED IN THE PROGRAM: A(1400,2)--THE MAIN DIAGONAL OF THE STIFFNESS MATRIX A(14oo,1)--THE OFF DIAGONAL OF THE STIFFNESS MATRIX DELTAHP(1400) - STORES CHANGES IN PRESSURES AND ELEV. DIFF(1400) — STORES PRESSURES FROM PREVIOUS ITERATIONS TO CHECK FOR CONVERGENCE EDIFF - UPPER LIMIT CONVERGENCE CRITERIA HEIGHT-—ELEVATION OF THE INPUT NODE HIN—-PRESSURE OF THE INPUT NODE HINP - PUMP HEAD ITERDELTA - DIFFERENCE IN PRESSURES BETWEEN ITERATIONS COVERGENCE CRITERIA ITS - NUMBER OF ITERATIONS KS - SPRINKLER CONSTANT LATC - LATERAL FRICTION CONSTANT LATD - LATERAL DIAMETER LATK - LATERAL CONSTANT LATM - LATERAL EXPONENT LATS - LATERAL SPACING M - SUBMAIN EXPONENT NEL — NUMBER OF SPRINKLERS PER LATERAL NL - NUMBER OF LATERALS P(1400) - STORES THE JUNCTION NODES OF THE SPRINKLER IRRIGATION SYSTEM RHS(700) - STORES THE VALUES OF THE RIGHT HAND SIDE OF THE SOLUTION EQUATIONS 0(1400) - STORES THE CHANGES IN FLOW RATE USED TO CALCULATED THE COMPONENT CONTRIBUTION 120 121 SLOPS - SUBMAIN SLOPE SLOPL - LATERAL SLOPE SPS - SPRINKLER SPACING SUBC - SUBMAIN FRICTION CONSTANT SUBD - SUBMAIN DIAMETER SUBK - SUBMAIN CONSTANT UNK - NUMBER OF UNKNOWS XS - SPRINKLER EXPONENT Y(1400) - STORES THE SPRINKLER FLOWS *****************************************************} {VARIABLE DECLARATION SECTION} CONST ARRAY SIZE = 700; TYPE VAR ARRAY:SIZEZ = 1400; ARRAY_TYPE = ARRAY[1..ARRAY_SI2E2] OF REAL; FILENAME,FILENAM:STRING[20]; LSST,INFILE:TEXT; A:ARRAY[1..ARRAY_SIZE,1..2] OF REAL; DELTAHP,DIFF,P,Y,Q:ARRAY_TYPE; ITERDELTA,EDIFF:REAL; ITS,UNK,UNK1:INTEGER; RHS:ARRAY[1..ARRAY_SIZE] OF REAL; KS,XS:REAL; TK,TX:REAL; {TEMPORARIES} KLAT,KSUB,KLAT1,KSUBI:REAL; SLOPLL,SPS:REAL; LATS,LATC,LATD,LATK,LATM,SLOPL:REAL; NL,NEL,NEL1,NSP,NLP:INTEGER; TSLOPS,TSLOPLL:INTEGER; HEIGHT,HIN,HINP:REAL; SLOPS,SUBD,SUBC,SUBK:REAL; BIAS,MUNK,ID,MP,LNO,I,J,K,CH:INTEGER; PTEMP,CTEMP,CTEMP1,CTEMP3,EE4,CONSTM,M:REAL; KCOML,COMKL,DFL,KCOMS,COMKS,DFS:REAL; SUML,MEANL,SUBFLOW:REAL; NOL,OFFSET:INTEGER; {END THE DECLARATION OF VARIABLES} {****************************************************} {PROCEDURE TO OPEN A FILE TO READ DATA FROM} PROCEDURE OPENINFILE; VAR OK:CHAR; FLAGl:BOOLEAN; BEGIN FLAG1:=FALSE; REPEAT 122 WRITELN('THE FILE TO READ DATA FROM IS ?'); READLN(FILENAME); WRITELN('THE FILE TO READ DATA FROM 15 ',FILENAME ,‘2 (Y OR N E=EXIT)'); READLN(OK); CASE OK OF 'Y':BEGIN FLAG1:=TRUE; ASSIGN(INFILE,FILENAME); RESET(INFILE); IF EOF(INFILE) THEN BEGIN WRITELN('UNABLE TO OPEN FILE',FILENAME); FLAGl:=FALSE; END; END; 'N':FLAG1:=FALSE; 'E':BEGIN WRITELN('NO FILE OPENED PROCEDURE EXITED'); FLAG1:=TRUE; END; ELSE FLAGI:=FALSE; END;{CASE} UNTIL FLAGI; END; {END OPENINFILE} {************‘k*****************************************} {THIS PROCEDURE OPENS A TEXT FILE TO WRITE TO} PROCEDURE OPENLSST; VAR FLAG1:BOOLEAN; OK:CHAR; BEGIN FLAG1:=FALSE; REPEAT WRITELN('THE NAME OF THE TEXT FILE TO WRITE TO IS ? '); READLN(FILENAM); WRITELN('THE TEXT FILE TO BE WRITTEN TO IS ',FILENAM, ' ? (Y OR N E=EXIT)'); READLN(OK); CASE OK OF 'Y':BEGIN FLAG1:=TRUE; ASSIGN(LSST,FILENAM); . REWRITE(LSST); END; 'N':FLAG1:=FALSE; 'E':BEGIN WRITELN('NO FILE OPENED. PROCEDURE EXITED'); FLAG1:=TRUE; END; 123 ELSE FLAG1:=FALSE; END; {CASE} UNTIL FLAGl; END; {END OPENLSST} {******************************************************} {PROCEDURE SETDATA READS THE SUBMAIN, THE LATERALS AND THE SPRINKLERS INFORMATION FROM A DATA FILE} PROCEDURE SETDATA; BEGIN READLN(INFILE,HIN,HINP,HEIGHT); READLN(INFILE,NL,NEL); READLN(INFILE,SUBD,SUBK,SUBC,COMKS,M,TSLOPS); READLN(INFILE,LATD,LATK,LATC,LATM,LATS,COMKL, TSLOPLL); READLN(INFILE,SPS,KS,Xs); END; {END SET DATA} {******************************************************} {PROCEDURE PRINTDATA WRITES THE INPUT DATA TO A FILE} PROCEDURE PRINTDATA; BEGIN WRITELN(LSST,‘ INPUT PRESSURE (FT) = ',' ',HIN:7:3); WRITELN(LSST,' PUMP HEAD (FT) = ',' ',HINP:7:3); WRITELN(LSST,' INPUT HEIGHT (FT) = ',' ',HEIGHT:7:3); WRITELN(LSST,‘ NUMBER OF LATERALS = ',' ',NL); WRITELN(LSST,‘ '); WRITELN(LSST,‘ ----- SUBMAIN DATA ----- '); WRITELN(LSST,‘ '); WRITELN(LSST,‘ DIAMETER (IN) = ',' ',SUBD:5:2); WRITELN(LSST,‘ CONSTANT K = ',' ',SUBK:6:2); WRITELN(LSST,' H—W FRICTION COEFF. = ',' ',SUBC:5:1); WRITELN(LSST,‘ TEES--ELBOWS COEFF. = ',' ',COMKS:4:1); WRITELN(LSST,' EXPONENT M = ',' ',M:5:3); WRITELN(LSST,‘ SLOPE % = ',' ',TSLOPS); WRITELN(LSST,‘ '); WRITELN(LSST,‘ ------ LATERAL DATA ----- '); WRITELN(LSST,‘ '); WRITELN(LSST,‘ DIAMETER (IN) = ',' ',LATD:5:2); WRITELN(LSST,‘ CONSTANT K = ',' ',LATK:6:2); WRITELN(LSST,‘ H-W FIRCTION COEFF. = ',' ',LATC:5:1); WRITELN(LSST,' VALVES COEFF. = ',' ',COMKL:4:1); 124 WRITELN(LSST,‘ EXPONENT M = ',' ',LATM:5:3); WRITELN(LSST,’ SLOPE % = ',' ',TSLOPLL); WRITELN(LSST,' LAT SPACING (FT) = ',' ',LATS:4:1); WRITELN(LSST,‘ '); WRITELN(LSST,’ ----- SPRINKLER DATA---- '); WRITELN(LSST,' '); WRITELN(LSST,' SPRINKLER COEFF. = ',' ',KS:7:4); WRITELN(LSST,‘ SPRINKLER EXPONENT = ',' ',XS:3:1); WRITELN(LSST,’ SPRINKLER SPAC.(FT)= ',' ',SPS:4:1); WRITELN(LSST,‘ NO. SPRINKLERS/LAT = ',' ',NEL); WRITELN(LSST,‘ '); WRITELN(LSST,‘ '); END; {PRINTDATA} {******************************************************} {FUNCTION POWER RAISES BASE TO THE POWER PWR.} FUNCTION POWER(BASE,PWR : REAL) : REAL; BEGIN IF BASE>0.0 THEN POWER:=EXP(PWR*LN(BASE)) ELSE POWER:=0.0; END; {POWER} {******************************************************} {PROCEDURE INITIALIZATION OBTAINS NEEDED VALUES OF CONSTANTS AND INITIALIZES THE ARRAYS} PROCEDURE INITIALIZATION; VAR I,J:INTEGER; BEGIN SLOPS:=TSLOPS/100.0; SLOPLL:=TSLOPLL/100.0; KLAT:=(LATK*SPs)/(POWER(LATC,M)*POWER(LATD,4.871)); KSUB:=(SUBK*LATS)/(POWER(SUBC,M)*POWER(SUBD,4.871)); DFL:=LATD/12.; DFS:=SUBD/12.; 125 {INITIALIzATION OF THE ARRAYS} FOR I:=1 TO ARRAY_SIZE DO BEGIN RHSII]:=0.0; FOR J:=l TO 2 DO A[1,J]:=o.o; END; FOR I:=l TO ARRAY_SIZEZ DO BEGIN DIFF[I]:=0.0; END; ITERDELTA:=0.01; EDIFF:=0.0; UNK:=NEL*NL+NL; UNK1:=UNK+1; NELl:=NEL+l; CONSTM:=(1./M)-1.; END; {END INITIALIZATION} {******************************************************1» {REAL FUNCTION C CALCULATES THE CONSTANTS NEEDED TO CREATE THE DIRECT STIFFNES MATRIX} FUNCTION C(I,ID : INTEGER) : REAL; BEGIN {SUBMAIN CONSTANT AND COMPONENT CONSTANT} IF ID=1 THEN BEGIN KCOMS:=0.00000012498*COMKS*POWER(Ql[I],0.148) /POWER9DFS,4); KSUB1:=KSUB+KCOMS; C:=POWER(ABs(P[I+NEL1]-P[IJ),CONSTM)* POWER(KSUBl,-1./M); {LATERAL CONSTANT AND COMPONENT CONSTANT} 126 IF ID=4 THEN BEGIN KCOML:=0.00000012498*COMKL*POWER(QI[II,0.148) /POWER(DFL,4); KLAT1:=KLAT+KCOML; C:=POWER(ABs(P[I+1]-P[I]),CONSTM)* POWER(KLAT1,-1./M); {SPRINKLER CONSTANT} IF ID=5 THEN BEGIN C:=POWER(ABS(P[I]-P[I+UNK1]), (-1.+XS))*KS; END {IF} ‘0 END; {FUNCTION c} {******************************************************} {PROCEDURE INITIAL HEIGHTS WILL GENERATE THE HEIGHTS OF THE NODES BASED ON SLOPES AND ON THE INITIAL ELEVATION OF THE INPUT NODE} PROCEDURE INITIAL_HEIGHTS; VAR I,J,OFFSET,BIAS,MP:INTEGER; BEGIN {INITIAL HEIGHTS} FOR I:=l TO NL DO BEGIN {FIRST CALCULATE THE ELEVATION OF THE LATERAL—SUBMAIN JUNCTION NODE} OFFSET:=UNK-(I-1)*NEL1; P[OFFSET+UNK1]:=P[2*UNK1]+(SLOPS*I*LATs); BIAS:=OFFSET-1; {PROCESS EACH EMITTER ON THE LEFT LATERAL} FOR J:=1 TO NEL DO BEGIN P[BIAS+UNK1]:=P[OFFSET+UNK1]+(SLOPLL*J*SPs); BIAS:=BIAS-1 END;{FOR J} .__—.’r— 127 END;{FOR I} END;{INITIAL HEIGHTS} {******************************************************} {PROCEDURE INITIAL PRES WILL MAKE INITIAL ESTIMATES OF NODE PRESSURES IN ARRAY P, FROM P[1] THROUGH P[UNK]} VARIABLES { DHL---CHANGE IN PRESSURE FOR THE LATERAL DHS---CHANGE IN PRESSURE FOR THE SUBMAIN QL----FLOW IN THE LATERAL QS----FLOW IN THE SUBMAIN} PROCEDURE INITIAL_PRES; VAR OFFSET,I,J,MP:INTEGER; QL,QS,T1,T2,DHS,DHL:REAL; BEGIN {CALCULATE THE SUBMAIN FLOW AND DELTA H BASED ON INPUT PRESSURE. OR IN THE PUMP HEAD} QS:=KS*POWER(P[UNK1],XS)*NL*NEL; {PUMP HEAD} T1:=3.6679/POWER(SUBC,1.852); T2:=POWER(QS,1.852)/POWER(SUBD,4.871); DHS:=T1*T2*LATS*NL; (CALCULATE ORIGINAL PRESSURES STARTING WITH THE LATERAL CLOSEST TO THE NODE CONTAINIG THE INPUT VALUES} FOR I:=1 TO NL DO BEGIN OFFSET:=UNK-(I-1)*NEL1; P[OFFSET]:=P[UNK1]-DHS*(1—POWER(1-I/NL,2.852)) -SLOPS*I*LATS; {CALCULATE PRESSURES DOWN THE LATERAL LINE} 128 BIAS:=OFFSET—1; QL:=KS*POWER(P[OFFSET],Xs)*(NEL1); T1:=3.6679/POWER(LATC,1.852); T2:=POWER(QL,1.852)/POWER(LATD,4.871); DHL:=T1*T2*SPS*NEL; FOR J:=1 TO NEL DO BEGIN P[BIAS]:=P[OFFSET]—DHL*(1-POWER(1-J/NEL,2.852)) -SLOPLL*J*SPS; BIAS:=BIAS-1 END; { END LOOP J} END; { END LOOP 1} {CHECK FOR NEGATIVE PRESSURE ESTIMATES } FOR I:=1 TO UNK DO BEGIN IF P[I] <=0.0 THEN P[11:=0.00001; END; {PRINT THE VECTOR CONTAINIG THE INITIAL HEIGHTS AND PRESSURES ESTIMATEs) WRITELN(LSST,'INITIAL HEIGHTS (FT) INITIAL PRESSURES (FT)'); FOR MP:=1 TO UNKI DO BEGIN IF MP=UNK1 THEN PIUNK11:=HIN; WRITE(LSST,' ',P[MP+UNK1]:8:2,' ',P[MP]:8:2); WRITELN(LSST); END;{PRINTING} PlUNKI]:=HIN+HINP; END; {INITIAL_PRES PROCEDURE} {*****************************************************t* MAIN PROGRAM *****************************************************t} {THIS IS THE MAIN PROGRAM-- IT CALLS SEVERAL PROCEDURES} BEGIN 129 OPENINFILE; OPENLSST; SETDATA; PRINTDATA; INITIALIZATION; {DEFINE CONSTANTS TO BE USED IN THE PROGRAM} UNK:=NEL*NL+NL; UNK1:=UNK+1; NEL1:=NEL+1; CONSTM:=(1./M)-1.; P[UNK1]:=HIN+HINP; P[UNK1*2]:=HEIGHT; ITERDELTA:=0.01; NOL:=NEL*NL; SUML:=0.0; INITIAL_HEIGHTS; INITIAL_PRES; {CALCULATE THE FINAL PRESSURES ESTIMATES. CALCULATE THE CHANGES IN ELEVATIONS AND STORE THEM IN DELTAHP} FOR MP:=1 TO UNK1 DO BEGIN PTEMP:=P[UNK1+MP]; DELTAHP[UNK1+MP]:=HEIGHT-PTEMP; P[MP]:=P[MP]+PTEMP; DIFF[MP]:=P[MP]; END; {FOR} {ADD THE CHANGE IN ELEVATION TO THE ORIGINAL PRESSURE ESTIMATE} FOR I:=1 TO UNK DO YII]:=P[I]+DELTAHP[I+UNK1]; {ITERATE UNTIL THE SOLUTION CONVERGES (EDIFF<1.0)} CONSTM:=(l./M)-1.; EDIFF:=100.; ITS:=1; WRITELN(' SPRINKLER IRRIGATION DESIGN EMPLOYING FINITE ELEMENT FORMULATION'); 130 WRITELN( ' ************************‘k***************** ' ); WRITELN(' THE PROGRAM IS ITERATING UNTIL SOLUTION IS OBTAINED W/IN 0.01 FT.'); WRITELN(' '); WHILE EDIFF>ITERDELTA DO BEGIN WRITELN(' ITERATION ',ITS:2,' . '); WRITELN(LSST,‘ ITERATION ',ITS:2,' . '); {CREATE THE STIFFNES MATRIX} FOR I:=1 TO UNK DO A[I,2]:= FOR I:=1 TO UNK DO A[I,1]:= FOR I:=1 TO UNK DO BEGIN IF I=((I DIV NEL1)*NEL1)THEN 0.0 0.0 {THIS IS THE FINAL NODE} IF I=UNK THEN A[I,2]:=A[I,2]+POWER(ABs(P{I+11-P[I]),CONSTM)* POWER(KSUB,—1./M) {THIS IS A SUBMAIN NODE PLUS A COMPONENT (TEE OR ELBOW)I ELSE BEGIN CTEMP:=C(I.1): A[I,2]:=CTEMP+A[I,2); A[I,1]:=-CTEMP+A[I,1]; A[I+NEL1,2]:=CTEMP+A[I+NEL1,2]; END {IF} {THIS IS A LATERAL W/ A SPRINKLER} ELSE BEGIN ID:=4; CTEMP:=C(I,ID); A[I,2]:=CTEMP+C(I,ID+1)+A[I,2]; A[I,1]:=-CTEMP; A[I+l,2]:=CTEMP+A[I+1,2]; END; {ELSE} END; {FOR} { ...................................................... FINISH CREATING THE STIFFNESS MATRIX ...................................................... } {CALCULATE THE RIGHT HAND SIDE OF THE SYSTEM OF EQUATIONS} 131 FOR I:=l TO UNK DO BEGIN IF I=((I DIV NEL1)*NEL1) THEN {THIS IS A SUBMAIN NODE} IF I<>UNK THEN RHS[I]:=0.0 ELSE {THIS IS FOR THE FINAL NODE} RHs[I]:=P[UNK1]*(POWER(ABS(P[UNK1J-PlUNK]),CONSTM)* POWER(KSUB,-1./M)) ELSE BEGIN {THIS IS A LATERAL WITH A SPRINKLER NODE} ID:=5; RHS[I]:=P[I+UNK1]*C(I,ID); END; {IF} END; {FOR} {NOW SOLVE THE EQUATION AX=RHS. FIRST MAKE THE MATRIX INTO UPPER TRIANGULAR FORM} NSP:=NEL1+1; FOR J:=2 TO UNK DO BEGIN {EXTENDED BANDWIDHT. SUBMAIN NODES} IF J=NSP THEN BEGIN EE4:=ABs(A[J-1,1]); A[J+NEL,2]:=A[J+NEL,2]-SQR(EE4)/A[J-1,2]; RHs[J+NEL]:=RH5[J+NEL]—A[J-1,1]*RHS{J-1]/A[J-1,2]: NSP:=NSP+NEL1; END ELSE BEGIN 132 {FOR THE LEFT LATERAL} EE4:=ABS(A[J-1,1]); A[J,2]:=A[J,2]-SQR(EE4)/A[J-1,2]; RHle]:=RHs[J]-A[J-1,11*RHs[J-1]/A[J-1,2]; END; {IF} END; {FOR} {NOW DO BACKWARD SUBSTITUTION TO FIND THE SOLUTION SET IN VECTOR P[I]} i*******************************************************} NSP:=UNK-NEL1; NLP:=UNK-NEL1; MUNK:=UNK-1; PlUNK]:=RHS[UNK]/A[UNK,2]; BIAS:=MUNK; FOR I:=1 TO MUNK DO BEGIN IF (BIAS<>NSP) AND (BIAS<>NLP) THEN P[BIAS]:=(RHS[BIAS]-A[BIAS,1]*P[BIAS+1])/A[BIAS,2] ELSE IF BIAS=NSP THEN BEGIN P[BIAS]:=(RHS[BIAS]-A[BIAS,1]*P[BIAS+NEL1])/A[BIAS,2]; NSP:=NSP—NEL1 END ELSE BEGIN PIBIAsl:=(RHs[BIAs]-A{BIAS,1]*P[BIAS+NEL1)I/A[BIAS,2]; NLP:=NLP—NEL1 END; BIAS:=BIAS-l; END; {*******************************************************} {CHECK FOR NEGATIVE PRESSURE ESTIMATES} FOR I:=1 TO UNK DO BEGIN IF P[I]<=0.o THEN P[I]:=0.00001; END; 133 {CHECK FOR CONVERGENCE. HERE USE THE CRITERION THAT THE SUM OF THE TOTAL PRESSURE VARIATION IN ALL THE NODES HAVE CHANGED LESS THAN 0.01 FEET} EDIFF:=0.0; FOR I:=l T0 UNK DO BEGIN EDIF:=ABs(DIFF[I]-P[I]) + EDIFF; DIFFII]:=PIII; END; ITS:=ITS+1; CALCULATE THE CP'S AND THE CORRESPONDING ELEMENT FLOW TO BE USED As THE NEW ESTIMATES IN THE SUBMAIN AND LATERAL COMPONENTS} FOR I:= 1 TO UNK DO BEGIN IF I=((I DIV NEL1)*NEL1) THEN {THIS IS FOR THE FINAL NODE} IF I = UNK THEN QILI]:=POWER(ABS(P[I+1]—P[I]),CONSTM) *POWER(KSUB,-1./M)*(AB5(P{I+1]-P[I]) ELSE BEGIN Ql[I]:=POWER(ABS(P[I+NEL1)-P[I]),CONSTM) *POWER(KSUB,-1./M)*(ABs(P[I+NEL1]-P[IJ)) END ELSE BEGIN QI[I]:=POWER(ABS(P[I+1]-P[I]),CONSTM) *POWER(KLAT,-1./M)*(Ass(P[I+1]-P[I])) END; {IF} END; {FOR} END; {WHILE} {THE SOLUTION HAS CONVERGED. READJUST THE PRESSURES BY ADDING CHANGES IN HEIGHTS} 134 FOR I:=1 TO UNK1 DO BEGIN PlI]:=P[I]-P[UNK1+I]; IF P[I]<=0.0 THEN P[I]:=o.00001; END; {FOR} (DISPLAY THE FINAL PRESSURE ESTIMATES} WRITELN(LSST,'FINAL PRESSURES ADJUSTED WITH HEIGHTS ); WRITELN(LSST,‘ FEET KILOPASCALS') FOR I:= 1 TO UNK1 DO BEGIN IF I=UNK1 THEN P[UNK1]:=HIN; WRITE(LSST,' ',P[I]:8:2,' ',P[I]*2.9891:8:); WRITELN(LSST); END; PIUNK11:=HIN+HINP; {CALCULATE THE SPRINKLER DISCHARGES AND PLACE THEM IN THE Y MATRIX. DISPLAY THEM IN GPM} FOR I:=1 TO UNK DO BEGIN TK:=KS; TX:=XS; Y[I]:=TK*POWER(P[I],Tx); END; {FOR} {DETERMINE THE SUBMAIN FLOW} FOR I:=1 TO UNK DO BEGIN IF (I MOD NEL1) <>o THEN SUML:= SUML+Y[I]; END; MEANL:=SUML/NOL; SUBFLO :=NL*(NEL*MEANL); SUBFLPS:=SUBFLOW*0.06309; WRITELN(LSST,'SUBMAIN FLOW(GPM) =',SUBFLOW:8:2 ,' (L/S)',SUBFLPS:8:2); WRITELN(LSST); WRITELN(LSST,‘ SPRINKLER FLOWS'); WRITELN(LSST,‘ GPM L/s ); FOR I:=1 TO UNK DO BEGIN IF (I MOD NEL1)<>0 THEN WRITE(LSST,' ',Y[I]:8:2,' ',Y[I]*0.06309:8:2); END; CLOSE(LSST); END. {END PROGRAM SPIRR-FE} APPENDIX C LINEAR THEORY COMPUTER MODEL LINEAR THEORY COMPUTER PROGRAM 1. Introduction The linear theory model developed by Wood (1980) is a FORTRAN based computer program capable of carrying out regular simulations of steady state pressure and flow in pipe distribution systems. In addition, extended period simulations can be carried out which simulate the Operation of the system over a period of time. The basis of the program is a direct solution to the basic pipe system hydraulic equations using a linearization technique and sparse matrix methods to handle the nonlinear terms in the energy equations. The linear theory computer program is appropiate for use on most IBM compatible microcomputer with at least 256K of memory and Microsoft DOS Operating system. 2. Data Coding Guidelines The input requirements for the linear theory computer model under the regular simulation mode are contained in 9 files. All of the required parameters are readily available or may be easily prepared. This section will discuss in general each of the files and parameters. 135 II. III. IV. 136 System Data a) type of simulation (regular or EPS) b) flow units (CFS, GPM, MGD, or lit/sec) C) number of pipes d) number of junction nodes e) number of PRV's f) program options Label Cards cards that contain any desired information which will be used as a label for the computer output Pressure Regulating Valve (PRV) Data a) junction node number for PRV b) pipe number - downstream from PRV c) grade set by PRV Pipeline Data a) connecting nodes b) length C) diameter d) roughness e) 2 (minor loss coefficients) f) pump power (or key to read pump data) 9) grade (if this pipe connects a Fixed Grade Node) h) pipe number (if nonconsecutive numbering specified) VI. VII. VIII. IX. 137 Pump Data head-discharge data for three operating points Junction Node Data a) demand b) elevation C) junction node number Output Option a) key for full or limited output. b) number of junction nodes for summary of maximum and minimum pressures C) Number of pipes for limited output d) number of junction nodes for limited output e) additional data keying program options Pipe numbers for limited output option Juction node numbers for limited output option APPENDIX D MODELS SIMULATION DATA 138 Slope = 2% uphill Comp. Coeff. Kc = 0.0 Junction Pressures Junction Node System Finite Element Linear Theory Number Elevations Pressures Pressures meters (ft) KPa (ft) KPa (ft) "’"’S""""'§2TEE-ZESI"IE§TZS'iZiT§§i”’I§§T£5’i£1?§éi’ 4 23.77 (78) 137.50 (46.00) 137.50 (46.00) 6 22.56 (74) 149.81 (50.12) 149.78 (50.11) 8 21.34 (70) 162.54 (54.38) 162.52 (54.37) 10 20.12 (66) 175.86 (58.83) 175.85 (58.83) 12 18.90 (62) 189.97 (63.56) 189.95 (63.55) 14 17.68 (58) 205.07 (68.61) 205.05 (68.60) 16 16.46 (54) 221.38 (74.06) 221.37 (74.06) 17 15.24 (50) 239.13 (80.00) 239.13 (80.00) Sprinkler Finite Element Linear Theory Number Flow Flow liters/sec (gpm) liters/sec (gpm) "'"'I”""'"""§T§Z’i’§3T§33'""""”"§T§Z‘Z'éif553’ 3 5.80 ( 92.01) 5.81 ( 92.04) 5 6.06 ( 96.13) 6.07 ( 96.17) 7 6.32 (100.23) 6.33 (100.27) 9 6.58 (104.35) 6.59 (104.39) 11 6.85 (108.56) 6.85 (108.60) 13 7.12 (112.89) 7.13 (112.94) 15 7.41 (117.40) 7.41 (117.45) 139 Slope = 1% uphill Comp. Coeff. Kc = 0.0 Junction Pressures Finite Element Junction Node System Linear Theory Number Elevations Pressures Pressures meters (ft) KPa (ft) KPa (ft) "m5"-'""56?I3’IEEIWESTBE'ZEETQBIWE56T5§7§ETEST 4 19.51 (64) 176.16 (58.93) 176.15 (58.93) 6 18.90 (62) 182.59 (61.08) 182.57 (61.08) 8 18.29 (60) 189.54 (63.41) 189.54 (63.41) 10 17.68 (58) 197.21 (65.98) 197.19 (65.97) 12 17.07 (56) 205.80 (68.85) 205.80 (68.85) 14 16.46 (54) 215.50 (72.10) 215.48 (72.09) 16 15.85 (52) 226.54 (75.79) 226.54 (75.79) 17 15.24 (50) 239.13 (80.00) 239.13 (80.00) Sprinkler Flows Sprinkler Finite Element Linear Theory Number Flow Flow liters/sec (gpm) liters/sec (gpm) """I"""""”"333-665353"""mm's'TZYZIBSTEIT 3 6.59 (104.44) 6.59 (104.48) 5 6.71 (106.38) 6.71 (106.42) 7 6.84 (108.43) 6.84 (108.47) 9 6.98 (110.66) 6.98 (110.70) 11 7.14 (113.10) 7.14 (113.14) 13 7.31 (115.79) 7.31 (115.84) 15 7.49 (118.79) 7.50 (118.84) 140 Slope = 0% Comp. Coeff. Kc = 0.0 Junction Pressures Junction Node System Finite Element Linear Theory Number Elevations Pressures Pressures meters (ft) KPa (ft) KPa (ft) ""'§""""1§T£2766731333731333"’"51336731333- 4 15.24 (50) 214.88 (71.89) 214.85 (71.88) 6 15.24 (50) 215.43 (72.07) 215.42 (72.07) 8 15.24 (50) 216.61 (72.47) 216.59 (72.46) 10 15.24 (50) 218.62 (73.14) 218.59 (73.13) 12 15.24 (50) 221.68 (74.16) 221.67 (74.16) 14 15.24 (50) 225.97 (75.60) 225.97 (75.60) 16 15.24 (50) 231.72 (77.52) 231.71 (77.52) 17 15.24 (50) 239.13 (80.00) 239.13 (80.00) Sprinkler Flows Sprinkler Finite Element Linear Theory Number Flow Flow liters/sec (gpm) liters/sec (gpm) """1 """"""" XSS'ZIESTEET"""""7357115257 3 7.29 (115.62) 7.30 (115.66) 5 7.30 (115.77) 7.31 (115.82) 7 7.32 (116.10) 7.33 (116.14) 9 7.36 (116.65) 7.36 (116.69) 11 7.41 (117.48) 7.41 (117.52) 13 7.48 (118.64) 7.49 (118.68) 15 7.58 (120.17) 7.58 (120.22) 141 Slope Junction Pressures = 1% downhill. Comp. Coeff. KC = 0.0 Linear Theory Pressures KPa 259.42 253.62 248.30 243.70 240.05 237.57 236.46 236.91 239.13 (ft) (86.79) (84.85) (83.07) (81.53) (80.31) (79.48) (79.11) (79.26) (80.00) Linear Theory Flow liters/sec (gpm) (127.37) (125.91) (124.55) (123.36) (122.42) (121.77) (121.47) Junction Node System Finite Element Number Elevations Pressures meters (ft) KPa (ft) ""7""m'1638'?SEEMEESTZZ'GESEI """"""""" 4 10.97 (36) 253.54 (84.86) 6 ll 58 (38) 248.32 (83.08) 8 12.19 (40) 243.72 (81.54) 10 12.80 (42) 240.08 (80.32) 12 13.41 (44) 237.59 (79.48) 14 14.02 (46) 236.47 (79.11) 16 14.63 (48) 236.91 (79.26) 17 15.24 (50) 239.13 (80.00) Sprinkler Flows Sprinkler Finite Element Number Flow liters/sec (gpm) """1 """""""" 5:63-655}? """"""""""""" 3 7.94 (125.86) 5 7.85 (124.50) 7 7.78 (123.32) 9 7.72 (122.37) 11 7.68 (121.72) 13 7.66 (121.43) 15 7.67 (121.54) (121.59) 142 Slope = 2% downhill Comp. Coeff. KC = 0.0 Junction Pressures Junction Node System Finite Element Linear Theory Number Elevations Pressures Pressures meters (ft) KPa (ft) KPa (ft) "'"E""""'{257153""3BZTIB'ZISITEIMSBZIT($6138I 4 6.71 (22) 292.45 (97.84) 292.42 (97.83) 6 7.92 (26) 281.25 (94.09) 281.21 (94.08) 8 9.14 (30) 270.87 (90.62) 270.84 (90.61) 10 10.36 (34) 261.56 (87.50) 261.54 (87.50) 12 11.58 (38) 253.53 (84.82) 253.50 (84.81) 14 12.80 (42) 246.99 (82.63) 246.96 (82.62) 16 14.02 (46) 242.12 (81.00) 242.11 (81.00) 17 15.24 (50) 239.13 (80.00) 239.13 (80.00) Sprinkler Flows Sprinkler Finite Element Linear Theory Number Flow Flow liters/sec (gpm) liters/sec (gpm) """1 """"""" 83175535) """""" 5357135373" 3 8.54 (135.37) 8.54 (135.41) 5 8.37 (132.69) 8.37 (132.74) 7 8.21 (130.16) 8.21 (130.21) 9 8.07 (127.85) 8.07 (127.90) 11 7.94 (125.83) 7.94 (125.88) 13 7.83 (124.16) 7.84 (124.21) 15 7.75 (122.90) 7.76 (122.95) 143 Slope = 2% uphill Comp. Coeff. KC 8 1.0 Junction Pressures Junction Node Number Linear Theory Pressures KPa (ft) System Finite Element Elevations Pressures meters (ft) KPa (ft) 24.99 (82) 123.66 (41.37) 23.77 (78) 135.72 (45.41) 22.56 (74) 148.07 (49.54) 21.34 (70) 160.91 (53.83) 20.12 (66) 174.44 (58.36) 18.90 (62) 188.90 (63.20) 17.68 (58) 204.52 (68.42) 16.46 (54) 221.58 (74.13) 15.24 (50) 239.13 (80.00) Sprinkler Flows 123.69 (41.38) 135.73 (45.41) 148.08 (49.54) 160.93 (53.84) 174.44 (58.36) 188.91 (63.20) 204.51 (68.42) 221.58 (74.13) 239.13 (80.00) Sprinkler Number Finite Element Flow liters/sec (gpm) Linear Theory Flow liters/sec (gpm) ( 85.84) ( 90.02) ( 94.12) ( 98.21) (102.35) (106.61) (111.03) (115.67) ( 85.89) ( 90.07) ( 94.18) ( 98.27) (102.44) (106.66) (111.09) (115.73) 144 Slope = 1% uphill Comp. Coeff. Kc = 1.0 Junction Pressures Junction Node System Finite Element Linear Theory Number Elevations Pressures Pressures meters (ft) KPa (ft) KPa (ft) ""'S"'""'§6T15’iEEI"'EE§T5§'ZQETEEI'"‘iéiiéa’iééiiii 4 19.51 (64) 173.98 (58.21) 173.99 (58.21) 6 18.90 (62) 180.47 (60.38) 180.48 (60.38) 8 18.29 (60) 187.57 (62.75) 187.59 (62.76) 10 17.68 (58) 195.51 (65.41) 195.51 (65.41) 12 17.07 (56) 204.51 (68.42) 204.51 (68.42) 14 16.46 (54) 214.85 (71.88) 214.85 (71.88) 16 15.85 (52) 226.77 (75.87) 226.75 (75.86) 17 15.24 (50) 239.13 (80.00) 239.13 (80.00) Sprinkler Finite Element Linear Theory Number Flow Flow liters/sec (gpm) liters/sec (gpm) "“""I""’"’""""’3:SS’IIBBTSEI”’""’"""ETSE'ZEBST£EI' 3 6.45 (102.21) 6.45 (102.27) 5 6.57 (104.15) 6.57 (104.21) 7 6.70 (106.22) 6.71 (106.28) 9 6.85 (108.50) 6.85 (108.56) 11 7.00 (111.02) 7.01 (111.09) 13 7.18 (113.86) 7.19 (113.92) 15 7.38 (117.05) 7.39 (117.11) 145 Junction Pressures Junction Node System Finite Element Linear Theory Number Elevatio Pressures Pressures meters (f) KPa (ft) KPa (ft) "”5""""153;'IEBIMEIZIZ’GBT373"-31532756353 4 15.24 (50) 212.31 (71.03) 212.34 (71.04) 6 15.24 (50) 212.94 (71.24) 212.97 (71.25) 8 15.24 (50) 214.30 (71.69) 214.32 (71.70) 10 15.24 (50) 216.63 (72.47) 216.65 (72.48) 12 15.24 (50) 220.18 (73.66) 220.20 (73.67) 14 15.24 (50) 225.21 (75.34) 225.23 (75.35) 16 15.24 (50) 231.98 (77.61) 231.98 (77.61) 17 15.24 (50) 239.13 (80.00) 239.13 (80.00) Sprinkler Flows Sprinkler Finite Element Linear Theory Number Flow Flow liters/sec (gpm) liters/sec (gpm) "'-"1"'"m"'"XIIZIISTIEI""""""§TEIII13:13)" 3 7.14 (113.17) 7.14 (113.23) 5 7.15 (113.34) 7.16 (113.41) 7 7.17 (113.71) 7.18 (113.77) 9 7.21 (114.34) 7.22 (114.40) 11 7.27 (115.30) 7.28 (115.36) 13 7.36 (116.63) 7.36 (116.70) 15 7.47 (118.41) 7.47 (118.47) Slope = 0% Comp. Coeff. RC = 1.0 146 Slope = 1% downhill Comp. Coeff. Kc - 1.0 Junction Pressures Junction Node System Finite Element Linear Theory Number Elevations Pressures Pressures meters (ft) KPa (ft) KPa (ft) """ 3""-""163E7333"ZEN-(53363""5333;753:513' 4 10.97 (36) 250.69 (83.87) 250.72 (83.88) 6 11.58 (38) 245.46 (82.12) 245.49 (82.13) 8 12.19 (40) 241.07 (80.65) 241.10 (80.66) 10 12.80 (42) 237.79 (79.55) 237.81 (79.56) 12 13.41 (44) 235.89 (78.92) 235.90 (78.92) 14 14.02 (46) 235.61 (78.82) 235.63 (78.83) 16 14.63 (48) 237.20 (79.36) 237.21 (79.36) 17 15.24 (50) 239.13 (80.00) 239.13 (80.00) Sprinkler Flows Sprinkler Finite Element Linear Theory Number Flow Flow liters/sec (gpm) liters/sec (gpm) ""7"“"m""3:82-115;Eli""""""§TS§'ZIEET§Ei“ 3 7.77 (123.20) 7.78 (123.27) 5 7.69 (121.88) 7.69 (121.95) 7 7.62 (120.76) 7.62 (120.83) 9 7.57 (119.92) 7.57 (119.99) 11 7.53 (119.43) 7.54 (119.50) 13 7.53 (119.36) 7.53 (119.42) 15 7.56 (119.77) 7.56 (119.83) 147 Slope = 2% downhill Comp. Coeff. KC 8 1.0 Junction Pressures Junction Node System Finite Element Linear Theory Number Elevations Pressures Pressures meters (ft KPa (ft) KPa (ft) """S'"m"'ETZETIQIWSBS317166227"3663571662;I 4 6.71 (22) 289.10 (96.72) 289.13 (96.73) 6 7.92 (26) 278.01 (93.01) 278.04 (93.02) 8 9.14 (30) 267.87 (89.62) 267.91 (89.63) 10 10.36 (34) 258.99 (86.64) 259.00 (86.65) 12 11.58 (38) 251.62 (84.18) 251.62 (84.18) 14 12.80 (42) 246.02 (82.31) 246.03 (82.31) 16 14.02 (46) 242.44 (81.11) 242.44 (81.11) 17 15.24 (50) 239.13 (80.00) 239.13 (80.00) Sprinkler Flows Sprinkler Finite Element Linear Theory Number Flow Flow liters/sec (gpm) liters/sec (gpm) """1 """""""" 5337133333 """""" 532713331)" 3 8.36 (132.52) 8.37 (132.59) 5 8.19 (129.89) 8.20 (129.97) 7 8.04 (127.45) 8.05 (127.52) 9 7.90 (125.27) 7.91 (125.34) 11 7.79 (123.44) 7.79 (123.50) 13 7.70 (122.03) 7.70 (122.09) 15 7.64 (121.11) 7.64 (121.17) 148 Slope = 2% uphill Comp. Coeff. Kc = 2.0 Junction Pressures Junction Node System Finite Element Linear Theory Number Elevations Pressures Pressures meters (ft) KPa (ft) KPa (ft) """E""""§ZT§6'ZQ23"i1£iié§'iZBT§63""I§ET62’1Z6TESI 4 23.77 (78) 134.03 (44.84) 134.12 (44.87) 6 22.56 (74) 146.43 (48.99) 146.52 (49.02) 8 21.34 (70) 159.37 (53.32) 159.44 (53.34) 10 20.12 (66) 173.10 (57.91) 173.16 (57.93) 12 18.90 (62) 187.88 (62.85) 187.92 (62.87) 14 17.68 (58) 204.00 (68.25) 204.03 (68.26) 16 16.46 (54) 221.77 (74.19) 221.76 (74.19) 17 15.24 (50) 239.13 (80.00) 239.13 (80.00) Sprinkler Flows Sprinkler Finite Element Linear Theory Number Flow Flow liters/sec (gpm) liters/sec (gpm) """I"""”"""’ETSS'Z'QSTééi""""""ETSS'I'EZTSEI 3 5.56 ( 88.13) 5.57 ( 88.23) 5 5.82 ( 92.20) 5.82 ( 92.31) 7 6.07 ( 96.28) 6.08 ( 96.39) 9 6.34 (100.43) 6.34 (100.54) 11 6.61 (104.73) 6.61 (104.83) 13 6.89 (109.23) 6.90 (109.33) 15 7.19 (113.99) 7.20 (114.50) 149 Slope = 1% uphill Comp. Coeff. RC = 2.0 Junction Pressures Junction Node System Finite Element Linear Theory Number Elevations Pressures Pressures meters (ft) KPa (ft) KPa (ft) ""7""MEETISTEEIWIESf9573%?Eii'miééiéfiééiééi 4 19.51 (64) 171.92 (57.52) 172.02 (57.55) 6 18.90 (62) 178.47 (59.71) 178.57 (59.74) 8 18.29 (60) 185.70 (62.13) 185.80 (62.16) 10 17.68 (58) 193.88 (64.86) 193.96 (64.89) 12 17.07 (56) 203.29 (68.01) 203.35 (68.03) 14 16.46 (54) 214.22 (71.67) 214.26 (71.68) 16 15.85 (52) 226.99 (75.94) 226.99 (75.94) 17 15.24 (50) 239.13 (80.00) 239.13 (80.00) Sprinkler Flows Sprinkler Finite Element Linear Theory Number Flow Flow liters/sec (gpm) liters/sec (gpm) """1 """""""" 336735333 ”””””” 331735383" 3 6.31 (100.09) 6.32 (100.20) 5 6.44 (102.02) 6.44 (102.13) 7 6.57 (104.11) 6.58 (104.22) 9 6.71 (106.43) 6.72 (106.54) 11 6.88 (109.04) 6.89 (109.14) 13 7.07 (111.99) 7.07 (112.10) 15 7.28 (115.35) 7.28 (115.45) Junction Node Number 0GP 10 12 14 16 17 150 Junction Pressures System Elevations meters (ft) Finite Element Pressures KPa 209.69 209.88 210.58 212.10 214.73 218.76 224.49 232.22 239.13 (ft) (70.15) (70.21) (70.45) (70.96) (71.84) (73.19) (75.10) (77.69) (80.00) Slope = 0% Comp. Coeff. Kc = 2.0 Linear Theory Pressures KPa (ft) 209.80 210.01 210.70 212.22 214.82 218.83 224.54 232.22 239.13 (70.19) (70.26) (70.49) (71.00) (71.87) (73.21) (75.12) (77.69) (80.00) Sprinkler Number Finite Element Flow liters/sec (gpm) (110.78) (110.83) (111.02) (111.43) (112.13) (113.20) (114.70) (116.71) Linear Theory Flow liters/sec (gpm) (110.90) (110.95) (111.14) (111.55) (112.25) (113.31) (114.81) (116.80) Junction Node Number Sprinkler Number 151 Slope = 1% downhill Comp. Coeff. Kc = 2.0 Junction Pressures System Elevations meters (ft) 253.63 Finite Element Pressure KPa (ft) 247.88 242.74 238.55 235.62 234.27 234.79 237.48 239.13 (82.93) (81.21) (79.81) (78.83) (78.37) (78.55) (79.45) (80.00) Sprinkler Flows Finite Element Flow liters/sec (gpm) (122.08) (120.66) (119.38) (118.32) (117.58) (117.23) (117.36) (118.05) Linear Theory Pressures 253.77 (84.90) 248.03 242.89 238.68 235.75 234.37 234.82 237.45 239.13 (82.98) (81.26) (79.85) (78.87) (78.41) (78.56) (79.44) (80.00) Linear Theory Flow liters/sec (gpm) (122.22) (120.80) (119.51) (118.45) (117.70) (117.35) (117.47) (118.14) 152 Slope = 2% downhill Comp. Coeff. Kc = 2.0 Junction Pressures Junction Node System Finite Element Linear Theory Number Elevations Pressures Pressures meters (ft KPa (ft) KPa (ft) ""’§""""'§TZ§'ZESEMERE-853$}""5333673323E 4 6.71 (22) 285.92 (95.65) 286.08 (95.71) 6 7.92 (26) 274.93 (91.98) 275.11 (92.04) 8 9.14 (30) 265.03 (88.67) 265.19 (88.72) 10 10.36 (34) 256.55 (85.83) 256.67 (85.87) 12 11.58 (38) 249.81 (83.57) 249.92 (83.61) 14 12.80 (42) 245.11 (82.00) 245.16 (82.02) 16 14.02 (46) 242.74 (81.21) 242.74 (81.21) 17 15.24 (50) 239.13 (80.00) 239.13 (80.00) Sprinkler Flows Sprinkler Finite Element Linear Theory Number Flow Flow liters/sec (gpm) liters/sec (gpm) ””1 """""""""" 533713223} """""""" QTSYZISZEST 3 8.19 (129.79) 8.20 (129.94) 5 A 8.03 (127.22) 8.04 (127.37) 7 7.88 (124.86) 7.89 (125.00) 9 7.75 (122.80) 7.76 (122.94) 11 7.64 (121.14) 7.65 (121.27) 13 7.57 (119.97) 7.58 (120.08) 15 7.53 (119.38) 7.54 (119.47) 153 Slope = 2% uphill Comp. Coeff. Kc = 3.0 Junction Pressures Junction Node System Finite Element Linear Theory Number Elevations Pressures Pressures meters (ft) KPa (ft) KPa (ft) ""2 """" 5236752775633726353""ISBT§§726T333 4 23.77 (78) 132.44 (44.31) 132.62 (44.37) 6 22.56 (74) 144.87 (48.47) 145.06 (48.53) 8 21.34 (70) 157.91 (52.83) 158.09 (52.89) 10 20.12 (66) 171.82 (57.48) 171.96 (57.53) 12 18.90 (62) 186.91 (62.53) 187.03 (62.57) 14 17.68 (58) 203.50 (68.08) 203.56 (68.10) 16 16.46 (54) 221.95 (74.25) 221.94 (74.25) 17 15.24 (50) 239.13 (80.00) 239.13 (80.00) Sprinkler Flows Sprinkler Finite Element Linear Theory Number Flow Flow liters/sec (gpm) liters/sec (gpm) """I’m'"'"‘"ETIE’Z’ESTEST'm'm"$367353?" 3 5.45 ( 86.32) 5.46 ( 86.49) 5 5.70 ( 90.37) 5.71 ( 90.54) 7 5.96 ( 94.44) 5.97 ( 94.61) 9 6.22 ( 98.60) 6.23 ( 98.77) 11 6.49 (102.93) 6.50 (103.09) 13 6.78 (107.50) 6.79 (107.65) 15 7.09 (112.37) 7.10 (112.51) 154 Slope = 1% uphill Comp. Coeff. Kc = 3.0 Junction Pressures Junction Node System Finite Element Linear Theory Number Elevations Pressures Pressures meters (ft KPa (ft) KPa (ft) "”5-"""'SBTEE’EEEE"’IESTES"82:313-"7313775333 4 19.51 (64) 169.98 (56.87) 170.20 (56.94) 6 18.90 (62) 176.58 (59.07) 176.80 (59.15) 8 18.29 (60) 183.94 (61.54) 184.16 (61.61) 10 17.68 (58) 192.35 (64.35) 192.53 (64.41) 12 17.07 (56) 202.14 (67.62) 202.27 (67.67) 14 16.46 (54) 213.63 (71.47) 213.72 (71.50) 16 15.85 (52) 227.19 (76.01) 227.17 (76.00) 17 15.24 (50) 239.13 (80.00) 239.13 (80.00) Sprinkler Flows Sprinkler Finite Element Linear Theory Number Flow Flow liters/sec (gpm) liters/sec (gpm) """i"""m""ETBYZ'SETEST""""mETBQ'Z'SETZET 3 6.19 ( 98.06) 6.20 ( 98.25) 5 6.31 ( 99.98) 6.32 (100.18) 7 6.44 (102.09) 6.45 (102.28) 9 6.59 (104.45) 6.60 (104.63) 11 6.76 (107.13) 6.77 (107.31) 13 6.95 (110.20) 6.96 (110.36) 15 7.17 (113.72) 7.18 (113.86) 155 Slope = 0% Comp. Coeff. Kc = 3.0 Junction Pressures Junction Node System Finite Element Linear Theory Number Elevations Pressures Pressures meters (ft) KPa (ft) KPa (ft) "“3"”""1332"?E63”’563T39733T353“”5632;733:353 4 15.24 (50) 207.58 (69.45) 207.86 (69.54) 6 15.24 (50) 208.35 (69.70) 208.61 (69.79) 8 15.24 (50) 210.03 (70.26) 210.28 (70.35) 10 15.24 (50) 212.94 (71.24) 213.15 (71.31) 12 15.24 (50) 217.42 (72.74) 217.57 (72.79) 14 15.24 (50) 223.81 (74.87) 223.80 (74.90) 16 15.24 (50) 232.46 (77.77) 232.43 (77.76) 17 15.24 (50) 239.13 (80.00) 239.13 (80.00) Sprinkler Flows Sprinkler Finite Element Linear Theory Number Flow Flow liters/sec (gpm) liters/sec (gpm) 'm"1""""""'“ESE"ZIBETEZT'”""WETEE’HBET$33 3 6.85 (108.59) 6.86 (108.81) 5 6.86 (108.80) 6.88 (109.01) 7 6.89 (109.24) 6.91 (109.45) 9 6.94 (110.02) 6.95 (110.22) 11 7.02 (111.19) 7.03 (111.38) 13 7.12 (112.85) 7.13 (113.02) 15 7.26 (115.05) 7.27 (115.20) Junction Node Number 156 Slope = 1% downhill Comp. Coeff. Kc - 3.0 Junction Pressures System Elevations meters (ft) Finite Element Pressures KPa (ft) 250.96 245.23 240.17 236.17 233.57 232.74 234.01 237.74 239.13 (82.04) (80.35) (79.01) (78.14) (77.86) (78.29) (79.53) (80.00) Linear Theory Pressures KPa (ft) 251.29 245.55 240.47 236.46 233.83 232.94 234.10 237.72 239.13 (82.15) (80.45) (79.11) (78.23) (77.93) (78.32) (79.53) (80.00) Sprinkler Number Finite Element Flow liters/sec (gpm) 7.55 7.46 7.38 7.32 7.28 7.26 7.28 7.34 (119.64) (118.24) (116.99) (115.99) (115.34) (115.12) (115.45) (116.38) Linear Theory Flow liters/sec (gpm) (119.88) (118.47) (117.22) (116.21) (115.55) (115.32) (115.62) (116.53) 157 Slope - 2% downhill Comp. Coef. Kc = 3.0 Junction Pressures Junction Node System Finite Element Linear Theory Number Elevations Pressures Pressures meters (ft) KPa (ft) KPa (ft) """S'""""§Ti§7I53""5233735333"""SSZTSE’ZSQTEQE 4 6.71 (22) 282.92 (94.65) 283.27 (94.77) 6 7.92 (26) 272.03 (91.01) 272.39 (91.13) 8 9.14 (30) 262.34 (87.77) 262.68 (87.88) 10 10.36 (34) 254.24 (85.06) 254.52 (85.15) 12 11.58 (38) 248.09 (83.00) 248.30 (83.07) 14 12.80 (42) 244.24 (81.71) 244.36 (81.75) 16 14.02 (46) 243.03 (81.30) 243.01 (81.30) 17 15.24 (50) 239.13 (80.00) 239.13 (80.00) Sprinkler Flows Sprinkler Finite Element Linear Theory Number Flow Flow liters/sec (gpm) liters/sec (gpm) """1 """"""" QTIE'ZIESTEU """""" 33171367163" 3 8.02 (127.19) 8.04 (127.44) 5 7.87 (124.67) 7.88 (124.91) 7 7.72 (122.38) 7.74 (122.62) 9 7.60 (120.44) 7.61 (120.66) 11 7.50 (118.94) 7.52 (119.15) 13 7.44 (117.99) 7.46 (118.18) 15 7.43 (117.70) 7.43 (117.84) Junction Node Number 158 Slope = 2% uphill Comp. Coeff. Kc = 3.0 Pump Head = Junction Pressures System Elevations meters (ft) Finite Element Pressures KPa (ft) 236.60 249.41 263.27 278.61 295.90 315.63 338.34 239.13 (79.15) (83.44) (88.08) (93.21) (98.99) (105.59) (113.19) (80.00) 119.6 KPa Linear Theory Pressures KPa (ft) 249.56 263.40 278.71 295.92 315.56 338.13 239.13 (79.21) (83.49) (88.12) (93.24) (99.00) (105.57) (113.12) (80.00) Sprinkler Number Finite Element Flow liters/sec (gpm) 7.96 8.21 8.49 8.79 (113.01) (116.14) (119.26) (122.60) (126.20) (130.14) (134.50) (139.35) Linear Theory Flow liters/sec (gpm) (113.20) (116.29) (119.46) (122.80) (126.39) (130.32) (134.66) (139.49) 159 Slope = 1% uphill Comp. Coeff. Kc = 1.0 Pump Head = 119.6 KPa Junction Pressures Junction Node System Finite Element Linear Theory Number Elevations Pressures Pressures meters (ft) KPa (ft) KPa (ft) """ 5""""EBTEE'ZEE§"'E§;T§I'?§ET323””’5§£T23’Z§IT5£3 4 19.51 (64) 280.71 (93.91) 280.68 (93.90) 6 18.90 (62) 287.50 (96.18) 287.46 (96.17) 8 18.29 (60) 295.25 (98.78) 295.21 (98.76) 10 17.68 (58) 304.31 (101.81) 304.26 (101.79) 12 17.07 (56) 315.03 (105.39) 314.99 (105.38) 14 16.46 (54) 327.78 (109.66) 327.82 (109.67) 16 15.85 (52) 342.95 (114.73) 343.12 (114.79) 17 15.24 (50) 239.13 (80.00) 239.13 (80.00) Sprinkler Flows Sprinkler Finite Element Linear Theory Number Flow Flow liters/sec (gpm) liters/sec (gpm) """1 """"""" 5T1£’Zi§§?6§3 """""" ETIE'ZIESTIIK' 3 8.24 (130.54) 8.24 (130.59) 5 8.34 (132.14) 8.34 (132.20) 7 8.45 (133.95) 8.45 (134.01) 9 8.58 (136.03) 8.59 (136.10) 11 8.74 (138.46) 8.74 (138.53) 13 8.91 (141.30) 8.92 (141.38) 15 9.12 (144.60) 9.13 (144.72) 160 Slope = 0% Comp. Coeff. Kc = 0.0 Pump Head = 119.6 KPa Junction Pressures Junction Node System Finite Element Linear Theory Number Elevations Pressures Pressures meters (ft KPa (ft) KPa (ft) ""3""—"”£2327;637355:???1693;735:3336};3 4 15.24 (50) 323.02 (108.07) 323.00 (108.06) 6 15.24 (50) 323.84 (108.34) 323.81 (108.33) 8 15.24 (50) 325.57 (108.92) 325.55 (108.91) 10 15.24 (50) 328.53 (109.91) 328.50 (109.90) 12 15.24 (50) 333.02 (111.41) 333.02 (111.41) 14 15.24 (50) 339.34 (113.53) 339.36 (113.53) 16 15.24 (50) 347.79 (116.35) 347.84 (116.37) 17 15.24 (50) 239.13 ( 80.00) 239.13 ( 80.00) Sprinkler'Flows Sprinkler Finite Element Linear Theory Number Flow Flow liters/sec (gpm) liters/sec (gpm) """1 """"""" 3357133353 """"""" 5357123]? 3 8.99 (142.43) 8.99 (142.48) 5 9.00 (142.61) 9.00 (142.66) 7 9.02 (143.00) 9.03 (143.05) 9 9.06 (143.66) 9.07 (143.72) 11 9.13 (144.67) 9.13 (144.72) 13 9.22 (146.06) 9.22 (146.13) 15 9.33 (147.91) 9.34 (147.99) 161 Slope = 2% downhill Comp. Coeff. Kc = 2.0 Pump Head = 119.6 KPa Junction Pressures Junction Node System Finite Element Linear Theory Number Elevations Pressures Pressures meters (ft) KPa (ft) KPa (ft) "”5""""EDIE-Z153-"£63T18713ZT5ET"ZBSTSE’GSZSE) 4 6.71 (22) 391.50 (130.98) 391.67 (131.03) 6 7.92 (26) 380.85 (127.41) 380.99 (127.46) 8 9.14 (30) 371.68 (124.34) 371.79 (124.38) 10 10.36 (34) 364.46 (121.93) 364.55 (121.96) 12 11.58 (38) 359.66 (120.32) 359.71 (120.34) 14 12.80 (42) 357.72 (119.67) 357.68 (119.66) 16 14.02 (46) 359.09 (120.13) 358.96 (120.09) 17 15.24 (50) 239.13 ( 80.00) 239.13 ( 80.00) Sprinkler Flows Sprinkler Finite Element Linear Theory Number Flow Flow liters/sec (gpm) liters/sec (gpm) mmI’""""-""3T53’ZIEZTEQE""""""33571923? 3 9.61 (152.39) 9.62 (152.54) 5 9.48 (150.26) 9.49 (150.41) 7 9.36 (148.40) 9.37 (148.55) 9 9.27 (146.92) 9.28 (147.06) 11 9.21 (145.93) 9.21 (146.06) 13 9.18 (145.53) 9.19 (145.64) 15 9.20 (145.81) 9.20 (145.90) LIST OF REFERENCES LIST OF REFERENCES Adams, R.W. 1961. 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