analyzing water distribution systems for swine growing

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Computersand electronicsELSEVIERL in agricultureComputers and Electronics in Agriculture 13 (1995) 75-86

Analyzing water distribution systems for swine growingand finishing unitsRabi H. Mohtar a,*, Larry J. Segerlind b, Howard L. Person b

a Department of Agricultural and Biolo gical Engineering, The Pennsylvania State Universiry,University Park, PA 16802, USA

b Department of Agricultural Engineeting, Michigan State University, East Lansing, MI 48824, USAAccepted 10 March 1995

AbstractAn analysisprocedure for livestock water distribution systems with nipple waterersis presented. The procedure uses the finite element analysis techniques to calculate thepressuredistribulion in a systemwith severalbranchesand nipple waterers.The procedurealso optimize the system or flow uniformity and cost. The formulation allows the use of

the drop inlet as a dischargepoint to reduce the size of the solution matrix. The dischargeparameterswere developedat this inlet usingwaterer flow characteristics.The flow rate ateach nipple waterer can be then calculatedbasedon the drop inlet pressure.The solutionfor a typical network is presented.The method can be usedas he basis or the designandevaluation of water distribution systemsn swinegrowingand finishingunits.Keywords: Pipenetwork design and analysis;Water distribution systems;Nipple waterer;Livestock housingwater systems.

1. IntroductionA water distribution system is an integral part of livestock housing. Deliveringadequate quantities of water is a fundamental requirement for providing optimalenvironmental conditions for animal housed in confinement facilities. Water stresscan include significant losses in the meat production as well as losing immunityagainst certain diseases. Celis (1988) found that the rate of gain, feed intake andfeed conversion of growing pigs were significantly decreased if water flow was below70 ml/min. Consequently, It is the responsibility of the designer to provide theadequate water flow rate to all animals all the time at low cost. Significant savings

* Corres ponding author. Fax: (814) 863-1031

0168-1699/95/$09.50 0 1995 Elsevier Scien ce B.V. All rights reserved.SSDI 0168-1699(95)00016-X


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76 R.H. Mohtar et al. I Computers and Electro nics in Agriculture 13 (1995) 75-86

in initial system and operating cost are achieved by proper system design. Highflow uniformity increases the production by reducing the water stress. Uniformwater distribution becomes an important issue. In order to maintain the minimumadequate water supply in the entire livestock housing, some discharge outlets willhave higher than the minimum. In pipe network design, there is a trade-off betweenflow uniformity and system capacity. For a water supply of constant flow rateand pressure head, as the pipe sizes increase, the uniformity increases due to thereduced friction losses in the pipes. On the other hand, larger pipe sizes increase theinitial system cost and therefore reduce profitability. Proper designs for livestockwater distribution systems will reduce the sum of initial fixed and maintenance costof the water distribution and ensure high uniformity of flow. This creates a need fora procedure that allows engineers to adequately design livestock watering systems.

Water distribution systems are presently analyzed and designed using the finiteelement method, Bralts and Segerlind (1985) Haghighi et al. (1989, 1992) andMohtar et al. (1991). These analysis and design procedures are applicable tofarmstead water systems but their application for this type of water systems has notyet been presented.The objective of the study is to develop an analysis procedure to assist indesigning water distribution systems that include nipple waterers commonly used inswine growing and finishing units. The procedure should allow the designer to checkflow uniformity upon changing any system parameter and also evaluate alreadyexisting system for their water flow uniformity and adequacy.

Fig. 1 represents a typical layout of a livestock water distribution system. Itcomprises of a water source, main, submains, laterals, and drop inlets. A demandnode is defined as a node where water is being discharged. Each of these demandnodes represents a drop inlet. The pipe segment connecting two nodes is defined as

Submain Lateral

Fig. 1. Typic al livestock water distribution layout.


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R.H. Mohtar et al. /Computers and Electro nics in Agriculture 13 (1995) 75-86 771 Lateral

0

Fig. 2. Typical drop unit. Bracketed numbers refer to element numbers while, non-bracketed numbersrefer to node numbers.

Water SourceDemand NodeJunction Node

Fig. 3. Example network layout. Bracketed numbers refer to element numbers while, non-bracketednumbers refer to node numbers.

an element. Fig. 2 represents a drop inlet feeding four nipple waterers. Fig. 3 is anexample of a livestock farm distribution network. Solid nodes represent drop inlet(demand nodes) and hollow nodes are junction nodes (non-demand nodes). Thesample network solutions will use the examples of Figs. 2 and 3 but the procedure isgeneral and can be implemented on any layout.2. Theoretical development2.1. Finite element ormulation forpressure head distribution

The element matrices were derived from the basic pipe flow energy equationsto a standard finite element form. Standard finite element procedure was thenemployed to solve for the pressure distribution in the network. This section presents


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78 R.H. Mohtar et al. I Computers and Electro nics in Agrkulture 13 (1995) 75-86

a summary of this development. The pressure distribution in any hydraulic pipenetwork is governed by the Bernoullis equation for mechanical energy conservation.Applying this equation between any two nodes i and j in the network as upstreamand downstream nodes respectively yields

2Zi+H~+$+h,=Z,+Hj+~+hl (1)

where i and j are any two nodes or junctions in the flow network, Zi and Zj areelevations at nodes i and j (L), Hi and Hj are pressure heads at nodes i and j (L),Vi and V, are velocities (L/T) at nodes i and j respectively, h, is the mechanicalenergy input along path ij (L), hl is the frictional head losses along path ij (L), g isthe gravitational acceleration (LD?).

The general energy equation for computing head losses in a pipe network, hi wasdeveloped by Mohtar et al. (1991) for all the components of the water distributionnetwork of Fig. 1 and is written as:hl = k,Qm (2)

where h, is the head loss or gain (L) through the pipe network components: straightpipe, pipe fitting, pump, or nozzle, kr is a component specific coefficient, m is acomponent specific exponent, and Q is the volumetric flow rate (L3/I). Values for k,and m for different components were given by Mohtar et al. (1991).Substituting (2) in (1) and using the flow continuity equation, Q = Ai Vi, where

Ai is the cross-sectional area (L2) of the pipe at node i and Vi is the flow velocity(L/T) at node i, yield(Zi - Z,;) + (Hi - Hj) = klQm - $(VF - Vy) -h,

orQ = (Zi - Zj)C + (Hi - Hj)C (4)

whereC= 1

k,Q--%+&-hm 2gA; %A; Qis a linearizing coefficient.

For any element [e} connecting nodes i and j, and by defining flow coming into anode as negative while f low leaving a node as positive, the following nodal continuityequations for mass conservation could be written as


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-Qin, + Qout,= 0 (5)-Qin, + Qautj = 0where Qin; and Qinj and are incoming flow to nodes i and j, Q,,,; and Qou, areflows leaving nodes i and j.

The contributions of element {e} to (5) are Qouti and Qoutj which will be labeledQ! and Q!l in later equations for consistency with the literature. Writing elementcontribution to the above nodal equations and substituting for the flow as in (4) andwriting the results in a matrix form yield

(6)where Q!e is the flow contribution of element {e} to node i (L3/T), Qy is theflow contribution of element (e} to node j (L3/T), AZ is the change in elevation(Zi - .Zi) (L), _, -, is the element stiffness matrix,[ 1 is the element forcevector, is the unknown nodal pressure head, C is the element coefficient fromthe linearization of the energy (4).

Eq. (6) has the standard finite element form. It is derived for a single elementand can be written for all elements in the system. The set of element equations isarranged to form a global system of equations using the direct stiffness procedure(Segerlind, 1984). Direct stiffness algorithm yields a system of equations which havethe general matrix form

(7)where {H} is the unknown nodal pressure head vector, [K] is the global stiffnessmatrix, {F] is the global force vector.The global system of equations in (7) is solved for the unknown heads at allnodes. For more details in the procedure and formulation, refer to Haghighi et al.(1989) and Mohtar et al. (1991).The nodal pressure head values computed in (7) are at nodes representing dropinlets feeding a series of the nipple waterer, Figs. 1 and 3. The pressure at each ofthe waterers in Fig. 2 is computed using the following relation

h = Hi - h (drop) + El (8)where h is the nodal pressure at the waterer (L) (nodes 7,8,9, and 10 in Fig. 2), Hiis the total head at drop inlet (L) (node 1 in Fig. 2, demand nodes of Figs. 1 and 3)hi (drop) s the sum of friction head losses of all elements from drop inlet node to thewaterers (L), El is drop length or the distance from drop inlet to the waterers, i.e.elevation difference between drop inlet and waterer.


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80 R.H. Mohtar et al. /Computers and Electro nics in Agriculture 13 (199.5) 75-862.2. Nodal flow distribution

The outflow from any discharge nozzle is dictated by the head at the dischargepoint. The nodal f low is related to the nodal head by the following relationq = kh; (9)

where q is the flow at node i (L3D), hi is the waterer head at node i (L), aand k are exponent and coefficient of discharge respectively and are determinedexperimentally.If the drop unit is treated as a discharge nozzle, applying the orifice flow (9) toany drop inlet node i in Fig. 1 yieldsQ=KHi (10)

where Q is the flow out of the drop inlet at node i, Hi is the head at dropinlet at node i (L), K is the drop coefficient of discharge, a is the drop exponentof discharge. Because the flow exponent changes only with the type of the flowresistance at the discharge nozzle, the exponent a is the same in (9) and (10).For a symmetrical drop with similar waterer characteristics, Q is related to asingle waterer discharge q by the following equationQ = nq

where IZ is the number of waterers per drop.(11)

Assuming that the sum (-hl (drop)+ El) in (8) is negligible compared to the inletpressure Hi, then hi and Hi are assumed equivalent. Therefore, for the drop unit inFig. 2 and for the series of n waterers, the K term of (10) could be written asK = nk (12)

The above assumption that hi and Hi are equivalent is only used to derive (12).Eq. (8) is used throughout the analysis to compute the pressure head at the watererfrom the pressure head at the drop inlet.The finite element formulation utilizes the orifice flow equation, (lo), whichsolves for the variable outflow as a function of the pressure head. The coefficientand the exponent of flow are considered constants for a given drop unit. Theformulation incorporates (10) into the analysis by treating the drop unit as adischarge point and as a separate element. The linearized discharge coefficient,c,,,,le, is added to the diagonal value of the stiffness matrix in (7), [Kii] where iis the node number, at the demand node. The linearized discharge coefficient is apositive non-zero linear function of the discharge coefficient, K, and exponent, a, of(lo), Mohtar et al. (1991). The value of cnozzle educe to zero for the nodes that donot connect to any drop units (non-demand nodes).3. Flow uniformity

The previous sections presented a procedure to determine the nodal pressureand flow distribution in a pipe network. The two parameters are related by the


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R.H. Mohtar et al. /Computers and Eiectr onics in Agriculture 13 (1995) 75-86 81Table 1Physical parameters for the typical drop in Fig. 2Element Node i Node j Description(1) I 2 4 Galvanized steel pipe(7-l 2 3 i Rubber pipe(3) 3 4 4 Galvanized steel pipe(4) 4 6 1 Galvanized pipe(5) 4 5 i Galvanized pipe(6) 5 I 4 Rubber pipe(7) 5 8 i Rubber pipe(8) 6 9 4 Rubber pipe(9) 6 10 g Rubber pipeNodes 7, 8, 9, and 10 have waterers.Table 2Physical parameters and equations for the example network of Fig. 3ParametersPipe length = 30.48 m (100 ft )Pipe size = SO.8 mm (2)Hazen-Williams coefficient, CR-w = ll.50

Elements(l)-(3), W(8), (lTj-(Wc (9, (lo), and (15) (L)-(3), (6)-(g), (If)-(13% 4% (Wand (15)(1)-(3X (6&(g); WbU%f% P), altall

Flow exponent (a) = 0.5Flow coefficient (k) = 1.33 em4 (for a single waterer)Equations:Hazen-Williams pipe frict ion: ht = 4 13 Q.852C-$2 D4.9871 Lwhere CH-w is a roughness coefficient, D is a pipe diameter (L)Orifice flow: 4 = kh.

orifice flow equations, (9) and (10). Uniform pressure distribution is needed toensure uniform flow at the nipple waterer. Excess pressure at the waterer leadsto a disruption of the uniformity and a high initial pipe network cost, Mohtar etal. (1991). Uniform flow in a livestock housing and many other agricultural pipenetworks is desirable and lead to a reduction in the tied cost of the system. In orderto maintain the minimum pressure needed for adequate flow at all nodes of thenetwork, some nodes will receive excess pressure. Uniformity can be achieved by aproper design and pipe size selection to consume this excess pressure. For a certainpressure range installing pressure compensating waterers can add to the uniformity.The major friction losses in a pipe network are generated in a straight pipesection, although losses through ofher components in a livestock water distributionnetwork can be significant. For a certain network layout, reducing the pipe sizesleads to :an increase in frictional losses (refer to the Hazen-Williams equation inTable 2). Mohtar et al. (1991) presented a formulation to design and optimize the


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82 R. H. Mohtar et al. I Computers and Electro nics in Agnculture 13 (1995) 75-86

network. The philosophy behind an optimized network is to deliver the requiredamount of water at no excess pressure. The approach was to trim down on pipe sizesand consume the excess pressure head in frictional losses. The same formulationwas used in the current study to design and optimize and re-size the network.3.1. Solution procedure

A comprehensive program for the design and optimization of water distributionnetworks, ANALYZER, was developed by Mohtar et al. (1991). The program usesthe finite element method as a numerical tool. ANALYZER was modified to includepeculiarities that are special to a swine housing water distribution system.The demand nodes in the network shown in Fig. 3 correspond to inlets to thedrop units. All the drop units are lumped into a nodal discharge at the drop inlet.Since all drops are usually identical and behave similarly, the output correspondingto Fig. 3 will be used for uniformity evaluations.

The global linear system (7) solves for the pressure distribution in the network.The flow in each element is calculated from the head values using (4) and (10).The problem is non-linear since the coefficients are functions of the unknown flow,therefore, the problem is solved iteratively and convergence is achieved within acertain prescribed tolerance. After the flow parameters were solved at the dropinlets, the pressure and flow at the discharge nozzle were traced and computedusing (8) and (9) respectively. Initial pipe sizes that will minimize the operating costof the system are determined. Excess pressure is calculated at each waterer andthe corresponding drop inlet is found. ANALYZER will try to down size the pipesleading to that inlet until the pipe size that satisfies the desired flow uniformityis found. Consequently, another set of iterations were needed re-size the networklinks, and insure flow uniformity. The critical path concept was used in the searchfor optimal pipe sizes. For a detailed description on the optimization procedurerefer to Kamand et al. (1987) and Mohtar et al. (1991).3.2. Example network

The example network in Fig. 3, with corresponding physical parameters inTable 2, was solved using the modified ANALYZER. Nodes 1, 2, 3, 6, 7, 8, 12, 13and 14 are nodes feeding drops similar to the one shown in Fig. 2. Tables 3 and 4show the results of the element flow rate and the nodal pressure for this examplenetwork.The distribution system layout and the waterer specifications, exponent andcoefficient of discharge and the number of waterers per drop, are user suppliedinput. Fig. 4 shows an experimentally determined response of waterer flow rate asaffected by inlet pressure, (Celis, 1988). The curves generated follow the orificeflow equation in (9). The parameters a and k in (9) were determined using anexponential regression fit to the experimental data points of Fig. 4. Results of theregression for a and k are shown in Table 2. These were used to evaluate the flow inthe network as described earlier.


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R.H. Mohtar et al. i Computers and Electronics in Agriculture 13 (1995) 75-86 83Table 3Results of element flow rates for the example network of Fig. 3Element Nodes Flow (I/s)23456789IO

III?131415

2 1 0.0793 2 0.1564 3 0.2385 4 0.23810 5 0.238I 6 0.0828 I 0.1679 8 0.25211 10 9 0.238 0.25216 11 0.49313 12 0.09914 13 0.20115 14 0.30617 16 15 0.493 0.30618 17 0.799

Table 4Results of nodal pressure heads for the example network of Fig. 3Node number Pressure head Cm)

1 8.132 8.133 8.964 10.375 10.396 9.03I 9.458 10.179 11.7510 11.8111 11.812 13.2613 13.85

14 14.8915 17.1416 17.21I7 17.2218 30.48Elevation is zero for all nodes.

The critical nodes 1, 6, and 12 in Fig. 3 have a total head of 8.13, 9.03 and 13.26m, respectively. These are defined as nodes with the highest pressure requirementfor a particular water flow path. Corresponding to that flow rates through criticalelements (1) (6) and (11) are 0.079, 0.082 and 0.099 l/s respectively. Assumingall nodes have the same flow and pressure requirements of 0.079 l/s at 8.13 m ofpressure head, these values correspond to 11% and 60% variation in pressure and


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0.0800 --

0.0666zP 0.0530?2 0.04005k 0.0266

PRESSURE, (kPa)Fig. 4. Pressure flow rate relations for the water used in Fig. 2

3.8% and 25.3% variation in flow at nodes 6 and 1 respectively compared to node1. The above situation is a major issue in uniformity evaluation and need to becorrected. Optimizing the above results means consuming 0.9 m of pressure headin the path from node 9 to node 6 and 5.13 m in the path from node 15 to node12. This process of trimming is performed using ANALYZER using the critical pathconcept and is described,by Mohtar et al. (1991). The pipe size availability and theprice per unit length are critical factors affecting the final pipe sizes in the network.Using waterers with adjustable setting (Celis, 1988), pressure compensatingwaterers, can also be used to improve flow uniformity. The problems with thisapproach are, first, there is no reduction in fixed costs, second, most waterers have anarrow pressure range where outflow is not affected by the inlet pressure.The pressure distribution within the drop unit is determined from the dropinlet pressure using (8). The frictional head losses hl for the waterer at node 10of Fig. 2 is the sum of frictional losses in elements (1) (2) (3) (4) and (9).Similarly, the frictional losses through paths leading to waterers at nodes 7, 8, and9 are determined by summing the frictional losses through the corresponding waterpath from the drop inlet to the waterer discharge point. For a symmetrical droplayout and similar waterer characteristics, the following equations are established tocompute the head loss in the drop of Fig. 2

W) hi181hi(drop) = hII11 + h1{2} + hl{3} + or + or

hI51 hU1(13)

or


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Table 5Results of element flow rates for the typical drop of Fig. 2Element Nodes Flow rate (l/s)1 1 2 0.122 2 3 0.123 3 4 0.124 4 6 0.065 4 5 0.066 5 7 0.03I 5 8 0.038 6 9 0.039 6 10 0.03

Table 6Results of nodal pressure heads for the typical drop of Fig. 2Node number Elevation (m) Pressure head (m) Total head (m)1 3.05 15.24 18.292 2.44 15.84 18.283 1.83 16.28 18.094 1.22 16.86 18.085 0.61 17.47 18.086 0.61 11.41 18.08I 0.0 17.56 17.568 0.0 17.56 17.56Y 0.0 17.56 17.5610 0.0 17.56 17.56

where hi(i) is the frictional losses through element i and determined from (2).Alternatively, the drop unit of Fig. 2 can be treated as a water distributionnetwork with a source pressure equal to the head at the drop inlet. Eq. (7) can besolved for the pressure at the various nodes of the drop unit. The typical drop ofFig. 2 with corresponding physical parameters in Table 1 was solved as a sampledrop example. Tables 5 and 6 show the element flow rate and nodal pressure for thetypical drop of Fig. 2. Similar output could be established as needed for the variousdrops.3.3. Features of the technique

The finite element method is a systematic numerical procedure for solvingcomplex engineering problems. The method has been implemented to solve thehydraulic system for a water distribution system used for various types of livestock.The presented technique was implemented on an IBM PC and is available in bothlanguages, Turbo Basic and FORTRAN. In addition to all the benefits of the finiteelement method discussed by Haghighi et al. (1989, 1992), the technique is flexiblein handling all livestock housing systems of various configurations and complexity.


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It will allow the user to change any of the design parameters, particularly pipediameter and input source pressure. The goal is to adequately supply water to allnipple waterers without any excess in pressure, since extra pressure is equivalentto extra operational cost. The technique is well suited to access optimizationsubroutines that will optimize system cost subjected to uniformity constraints.4. Summary and conclusions

A procedure for livestock water distribution systems analysis is developed andpresented using the finite element method. The element matrices were derived fromthe basic pipe flow energy equations. Standard finite element procedure was thenemployed to solve for the pressure distribution in the network. The finite elementformulation utilizes the orifice flow equation which solves for the variable outflowas a function of the pressure head. The coefficient and the exponent of flow arefixed for a given discharge outlet. A linearized discharge coefficient is added to thediagonal value of the stiffness matrix. The formulation is illustrated with practicalexamples and used in the design of a distribution network and a drop unit. Theformulation allows the use of the drop inlet as a discharge point to reduce the sizeof the solution matrix. The discharge parameters were developed at this inlet usingwaterer flow characteristics. The flow rate at each nipple waterer is calculated basedon the drop inlet pressure. In addition to the design, the procedure uses the criticalpath concept to optimizes the system cost and flow uniformity. The critical pathconcept is an iterative procedure that search for the critical path, sub-critical path,etc. At every iteration the excess pressure at each node are determined. Excesspressure will be utilized by re-sizing the off-critical path pipes. The procedure goeson until no more excess pressures exists. The method can be used as the basis forthe design of water distribution systems in swine growing and finishing units.ReferencesBralts, V.F. and Segerlind, L.J. (1985) Finite element analysis of drip irrigation units. Trans. ASAE ,

28(3): 809-814.Cehs, J.E. (1988) Effect of water flow rates on performance of nursery pigs and influence of pressure

on flow rate from nipple waterer. M.S. Th esis presented to Michigan State University, East Lansing,MI.

Haghighi, K., Bralts, J.F., Mohtar, R. and Segerlind, L.J. (1989) Modeling expansion/contraction, valveand booster pump in hydraulic pipe network analysis. A finite element approach. Trans. ASAE,32(6): 1945-1955.

Haghighi, K., Mohtar, R.H., Bralts, V.F. and Segerlind, L.J. (1992) A linear formulation model for pipenetwork componen ts. Comput. Electron. Agric., 7: 301-321.

Kamand, F.Z., Mohtar, R.H., and Baasiri, M. (1987) A model for the design and optimization of pipenetworks. ASA E Paper No. 87-2614, ASA E, St. Joseph, MI.

Mohtar, R.H., Bralts, V.F. and Shayya., W.H. (1991) A Finite element formulation for the analysis andoptimization of pipe networks. Trans. ASA E, 34(2): 394-401.

Segerlind, L.J. (1984) Applied Finite Element Analys is. John Wiley and Sons, New York.

analyzing water distribution systems for swine growing

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