FINITE ELEMENT METHOD ANALYSIS OF MICROIRRIGATION

SYSTEM PRESSURE DISTRIBUTION

KANG Yaohu*and Soichi NISHIYAMA*

* Faculty of Agriculture,Kagawa University

Abstract It is very important to estimate corrective pressure distribution to analyse uniformity

of water application in microirrigation system.Finite element method is one of the very convenient

and accurate methods to do this analysis.If the nodal system is arranged and simplified reasonably,

personal computer can be used to calculate pressure distributioneven in a system with a large

number of water application devices, such as line source emitterand high density point source emit-

ters of trickle irrigation system.The accuracy of the solution is very high because theoretical cal-

culating procedure is used.The convergence is quick no need to think whether the initial estimates

of the nodal pressure are near or far from the final solution.

I. INTRODUCTION

As it has been expounded by many resea-

chers and engineers that estimation of correc-

tive pressure distribution is very important for

designing microirrigation system to apply uni-

form water to the irrigation field.Nowadays,

there are many methods,such as Hardy Cross

method(Described by Chenoweth and Crawfor-

d,1974,and Jeppson,1977),Newton-Raphson

method,the linear theoretical method (Wood

and Charles,1972),and the nonlinear theoretical

method by using iterative techniques (Solomon

and Keller,1974,Wu and Fangmeier,1974,and

Perold,1977).Every method has its own advan-

tages and disadvantages.For example,Hardy

Cross method can be used to get accurate

result but its convergence is slow due to the in-

dependent solution of loop and nodal equation.

The convergence of Newton-Raphson method is

quick if the initial estimated value is near tothe final solution but sometimes it has not ten-

dency to get the solution when initial estimate

is quite far off.The common disadvantages of

the above mentioned methods are concentrated

on the hydraulic analysis of a single lateral

line.

The finite element method is a systematic

numerical procedure for solving a complex

engineering problems and it was applied in the

analysis of drip irrigation submain unit Bralts

and Segerlind in 1985.The convergence of this

method is very quick since it considers the

boundary condition.It is also possible to calcu-

late the pressure variation directly as far as

the pressure at each emitter is known.This

method can be used as it is mentioned above,

however,it needs large computer memory due

to large matrix equation which is not conve-

nient for calculating with personal computer.

In this paper,some mathematical models will

be applied to analyse pressure distribution in

various microirrigation systems with finite ele-

ment method more efficiently and some

methods will be presented to reduce nodal num-

ber rationally to make that it is possible to use

finite element method analysing pressure distri-

bution in a microirrigation system with a large

number of water application devices such as

line source emitter and high density point

source emitter,etc..

II. THE MODELLING OF FINITE

ELEMENT METHOD

As it is shown in Fig.1,the finite element

*

香川大学農学部

キ ー ワ ー ド:Finit eelement method,Microirrigation,

Pressure distribution

農土 論集(169) 19

農業土木学会論文集第169号

model(Zienkiewicz,0.C.,1971)in electrical

system based on the Ohm's law of the relation

between the currents entering the element at

the ends and the voltages as:

(1)

(2)

Eq.(1)and(2)can be expressed in matrix equa-

tion form by

or in standard form of matrix equation of finite

element method:

(3)

where,It,L=the currents from the end point i

to the end point j and from j to i respectively,

r=the resistance of the wire element ij,Vi,Vj=

the electric potential at the end point i and j

respectively,e=the signal of element,{I}e=the

currents through element,[K]e=the stiffness

matrix of element,and{V}e=the potential

matrix of element.

In Eq.(2)or(3),if the electric resistance re

and the electric potential Vi and Vj are known,

Fig.1 Network of electrical resistance

Fig.2 The finite elements of lateral line in micro

irrigation water distribution system

the electric current I and I;can be obtained

after solving Eq.(3).

The water flow in pipe is similar to the elec-

tric current in wire except that the friction fac-

tor of water flow in pipe is changed with

Reynolds number while the electric resistance is

constant when electric current flows through

wire.If set a coefficient Kp for pipe flow as the

factor 1/re in electric current entering the wire

and assume that emitter riser height is neglect-

ed,the principle mentioned above can be

applied in microirrigation water distribution

system as it is shown in Fig.2.The nodal equa-

tions can be written as:

(4)

(5)

(6)

(7)

where,Qbr,Qbm,Qbs=the discharge of element

r,m,and s in and out node b respectively,Ha,

Hb,Hc=the pressure head at point a,b,c

respectively,KPr,KeS,Kee the coefficients

that can be determined by analysing the

hydrualics of lateral line and emitters and Za,

Zb,Zc=the elevation head above the reference

surface.

According to the mass conservation princi-

ples the total inlet discharge of a node is equal

to total outlet discharge of the node.Therefore,

(8)

(9)

If there is no field slope,all nodal elevations

are the same.Substitute Eq.(3),(4)and(5)to

Eq.(8),the following equation can be obtained.

(10)

Substitute Eq.(5)and(7)to Eq.(9)the follow-

ing equation can be obtained.

(11)

If Ha is system operating pressure,those

equations mentioned above can be written in

matrix equation as follows

or in standard form of matrix equation of finite

20 Trans. JSIDRE Feb. 1994

FINITE ELEMENT METHOD ANALYSIS OF MICROIRRIGATION SYSTEM PRESSURE DISTRIBUTION

element method

(12)

where,[K]=the stiffness matrix of element,

{H}=the unknown pressure head matrix of ele-

ment and{F}=the known matrix of force.

III.HYDRAULICS OF LATERAL LINE

AND SUBMAIN

Emitters are pressure dissipating unit of mi-

croirrigation system.The relationship between

emitter discharge and operating pressure can

be described as follow

(13)

where,C=the coefficient of emitter discharge,

q=emitter discharge,H=the pressure head in

the lateral,and x=the exponent of H in emit-

ter discharge equation(x=1.0 for laminar emit-

ter,x=0.5 for orifice-type emitters,and x=0

for pressure compensating emitters*etc.).

Eq.(13)can be rearranged as

(14)

where

(15)

is a coefficient of emitter.

For a straight pipe element as shown in Fig.

3 the Bernoulli's equation can be applied as

(16)

where,Zi,Z;=the upstream and downstream

elevation respectively,Hi,Hj=the upstream

and downstream water head respectively,s=the

emitter spacing(or lateral spacing),D=the

diameter of laterals(or submain),V2/(2g)=the

velocity head in the lateral(or submain),and

f=the friction factor which can be determined

as follows:

(17)

(18)

Fig.3 Straight pipe element

Fig.4 Microirrigatim system (emitter with outriser)

(19)

where,Re=Reynolds number.

Since V=4Q/ƒÎD2,Eq.(16)can be rearranged as

(20)

or

(21)

(22)

Equation(15)and(22)are used to determine

the coefficient Kp and Ke respectively.

IV.APPLICATION OF THE FINITE

ELEMENT PRESSURE ANALY-

SIS METHOD IN MICROIRRI-

GATION SYSTEM

1.Emitters without Riser

Fig.4 shows one type of microirrigation later-

als with emitters layout.The emitters do not

have riser.In this case,the nodes can be num-

bered orderly as shown in this figure.

The matrix equation for calculating pressure

* Pressure compensating emitter:the emitter is

designed to change emitter flow path or opening by

emitterself when the operating pressure increases or

decreases and it's discharge is constant within a maxi-

mum and minimum operating pressure range.

農土論集(169) 21

農業土木学会論文集第169号

Fig.5 The calculating matrix of finite element in the case of Fig.4.

distribution in the system is written in Fig.5

and it shows that the stiffness matrix is a sym-

metry matrix.

2.Emitters with Riser(spray,bubbler,or

sprinkler irrigation system)

Fig.6 shows a layout of trickle irrigation sys-

tem laterals and emitters with riser.In this

case,there is water head loss due to friction of

riser and elevation of emitter.When the riser is

short such as in spray system or small size

sprinkler system,the friction loss of riser is

small compare to the water head loss due to

elevation of rised emitters.The nodal system

can be illustrated as in Fig.4 when it is the

same as in Fig.6.The emitter discharge equa-

tion is expressed as

(23)

where

(24)

is a coefficient of emitter with riser in which

h=the height of riser.

Using Eq.(23)and(24),the matrix equation

for calculating pressure distribution of emitters

can be written as in the case of emitter without

riser.Eq.(24)makes the calculating procedure

more simplified and this method is named as

simplify method in this paper.

This simplification reduces the number of

nodes almost by half and it would be easy to

Fig.6 Microirrigation system(emitter with riser)

calculate with personal computer.The result is

almost the same as in the case of not-simplified

one.Example of verification and comparison

will be shown later.

If the riser is high such as in large scale

sprinkler irrigation system,there is significant

water head loss due to friction except the

losses due to elevation of emitter riser.In this

case,it is necessary to add additional nodes at

the joint point of riser and lateral and this is

the general finite element method in microirri-

gation system which is named as not simplifiedmethod in this paper.The number of nodes as

in Fig.6 would be convenient to make com-

puter programme.

3.Line Source Emitters

In microirrigation system with line source

emitters,the emitter spacing is less than one

meter.If every water application device is

arranged as node,it may be difficult to calcu-

22 Trans.JSIDRE Feb.1994

FINITE ELEMENT METHOD ANALYSIS OF MICROIRRIGATION SYSTEM PRESSURE DISTRIBUTION

late with personal computer since it takes long

time or the memory may not be enough.It is

also not advisable to calculate the pressure

head loss at each point of water application

device even with in one meter length of node

spacing because the pressure change is

insignificant compare with other factors such as

rise and fall of field and temperature etc.

Therefore,it is possible to increase the node

spacing and reduce number of nodes.Fig.7(a～c) showthecalculatedpressureprofileof1,

(a) Pressure profile of pressure compensating emitter(x=0)

(b) Pressure profile of orifice type emitter(x=0.5)

(c) Pressure profile of laminar emitter(x=1.0)

Fig.7 Calculated pressure profile for line source emitter

2,and 4 m spacing of pressure compensating

emitter (exponent x=0),orifice emitter(x=0.5)

and laminar emitter(x=1.0)respectively.The

discharge of these emitters are the same(3.33

10-6m3/s).

Fig.7 (a•`c)show about the maximum calcu-

lating error of pressure profile for three types

of line source emitters.When the node spacing

is increased from 1 to 2m,the maximum error

is not more than 0.03m and it is not more than

0.09m when the node spacing is increased from

1 to 4m in 100m length of laterals(diameter=

0.021m).This maximum error is the accumula-

tion of errors from starting point to the end

point of laterals.The increase of node spacing

results increase of calculated pressure varia-

tion.Therefore,it is safe from uniformity of

water application design point of view.This

treatment reduces the number of nodes about

1/2 when node spacing is arranaged from 1m

to 2 m and the number of nodes is reduced

about 3/4 when node spacing is arranged from

1 to 4m.This makes possible that finite ele-

ment method is used to calculate pressure dis-

tribution with personal computer for line

source emitters.

V.CALCULATING PROCEDURE

As it is shown in Fig.4. the calculating proce-

dure is summarized as follow:

(1) Give a pressure estimate at each node(itdoes not matter to give the pressure estimates

the same as operating pressure of system,and

the convergence is almost the same as given

good pressure estimates).(2) Calculate discharge of emitters.

(3) Obtain If Ke from Eq.(15).

(4) Add the downstream discharges of emit-

ters in the lower reaches to obtain discharge of

lateral elements

(25)

where,Qpi,Qej=the discharge of the lateral ele-

ment in the upper reaches of the node i and

the discharge of the emitters in lower reaches

of node i.

(5) Add the downstream discharges of lat-

農土論集(169) 23

農業土木学会論文集第169号

Fig.8 Layout of small size sprinkler irrigatim system

erals in the lower reaches of the node to obtain

discharge of the submain elements.

(6) Obtain Reynolds number from Eq.(26).

(26)

where,v=kinematic viscosity of water.

(7) Obtain friction factor from Eq.(17),(18)

or(19).

(8) Obtain Kp from Eq.(22).

(9) Solve the matrix equation to obtain the

solution,if the obtained solution is not equal to

the estimated one in the given places of deci-

mals,make this solution as new estimates and

repeat the calculating procedure until the solu-

tion of matrix equation is equal to estimates in

the given places of decimals and this would be

the final solution.

Usually,the circulation of computer pro-

grame is three to five times when calculated

pressure is equal to estimated pressure in 3

places of decimals.

VI.ANALYSIS EXAMPLES:

Fig.8 shows a small size of sprinkler irriga-

tion system.The field conditions and physical

parameters are listed in Table 1 and the solu-

tion is in Table 2.

Figs.9-11 show the pressure profile of the

laterals,pressure distribution at emitters and

pressure profile of submain line respectively.

It is found that the pressure at part 1 is

higher than in part 2.This is because of pres-

sure gain and pressure loss due to slope of

Fig.9 Pressure profile of lateral

field.

It can also be found that pressure at each

emitter is lower than at the joint point of emit-

ter riser and laterals almost by 150cm.This

means that the pressure loss due to friction of

riser is very small.

Table 1 Field conditions and physical parameters

24 Trans.JSIDRE Feb.1994

FINITE ELEMENT METHOD ANALYSIS OF MICROIRRIGATION SYSTEM PRESSUREDISTRIBUTION

Table 2 Operating pressure and calculated pressure

Fig.10 Pressure distribution of emitters

Fig.11 Pressure profile of submain

Fig.12 Comparison of pressure distribution at emitters

using simplified and not-simplified methads

Fig.13 Comparison of pressure distribution at emitters

using simplified and not-simplified methods

Figs.12,13 show the pressure distribution of

emitters that was obtained by using simplified

and not-simplified methods.It is illustrated that

the pressure distribution obtained from both

methods is more or less the same.This result

proves that the simplified method is reasonable.

All of the calculated pressure mentioned

above were checked by using the Step by Step

農土論集(169) 25

農業土木学会論文集第169号

method and it is found that results of the finite

element method are completely the same as the

results of the Step by Step method.

VII.CONCLUSION:

Finite element method can be used to analyse

pressure distribution in microirrigation system.

The advantages of this method includes to

obtain accurate pressure distribution at the

pressure controlled unit since the mathematical

procedure is theoretical.The uniformity of

water application can be determined directly

because the pressure of each emitter is

obtained.The convergence is quick and no

need to think about whether the initial esti-

mates of nodal pressure is near or far from the

final solution.It is possible to use in microir-

rigation system which has large number of

emitters such as line source emitters,etc.with

personal computer.

REFERENCES

1) Bralts, V. F. and Segerlind, Li.: Finite element

analysis of drip irrigation submain unit, Trans.

of ASAE, V. 28, pp.809•`814(1985)

2) Bralts and D. M. Edwards, Wu,I. P.: Drip irriga-

tion design and evaluation based on the statisti-

cal uniformity concept, Advances in Irrigation,

Vol.4, Academic Press Inc., pp.67•`117(1987)

3) Bernuth, R. D. and Tonya Wilson: Friction fac-

tors for small diameter plastic pipes, Journal of

Hydraulic Engineering, ASCE, Vol.115, No.2, pp.

183•`192(1989)

4) Fu, L., Dong, W. C. and Zheng, Y. Q. etc.: Techni-

cal guidance for microirrigation engineering, Pub-lished by Electric Power and Water Conservancy

Press, China, (1987)5) Kamand, F. Z.: Hydraulic friction factors for

pipe flow, Journal of Irrigation and Drainage

Engineering, ASCE, Vol.114, No.2, pp.311•`323

(1988)

6) Kell, J. and Blieser, R. D.: Sprinkler and trickleirrigation, Published by Van Nostrand Reinhold,

New York(1990)7) Nakayama, F. S. and Bucks, D. A.: Trickle irri-

gation for crop production, Elsevier Science Pub-lication B. V. (1986)

8) Warrick, A. W. and Yitayew M.: Trickle lateral

hydraulics I: Analytical Solution, Journal of Irri-

gation and drainage Engineering, ASCE, Vol.114,

No.2, pp.281•`288(1988)

9) Wu, I. P., Saruwatari C. A. and Gitlin, H. M.:

Design of drip irrigation lateral length on uni-

form slope, Irrig. Sci. 4, pp.11.7•`135(1983)

10) Zienkiewicz, O. C.: The finite element Method in

engineering Science, Published by McGraw HillPublication Company Limited, London(1971)

〔1993.3.11.受 稿 〕

〔この研 究論文 に対 す る公 開の質疑 あ るいは討議(4,000

字以内,農 業 土木学会論文集編集委 員会 あて)は,1994

年8月24日 まで受付 け ます。〕

26 Tran5.JSIDRE Feb.1994

農業土木学会論文集 第169号 内容紹介

〔研究論文〕変動する降雨入力のもとでの降雨浸透現象

- 降雨浸透現象の研究(II)-

富永雅樹変動する降雨強度が浸透現象に与 える影響 を考察す る

ために実験 を行 った。 まず,総 降雨量が同 じで,降 雨強

度 と継続時間が異 なる降雨入力下 での長 さ5mの カ ラ

ム実験か らは,継 続時間が長 ければ,短 時間の豪雨の と

きよ り,降 雨終了時の浸潤前線が地盤中の深部 に到達す

る現象が観察 された。 つぎに,一 定 の休止期 間を置 いて

くりか えす方形波状 の降雨入力下 での実験では,2mの

関東 ローム土地盤では方形波状 の降雨入力に対応 した地

下水流出があること等 を観測 した。 これ らの現 象か ら,

動的平衡浸透状態での水分量は降雨入力の強度 の影響 を

受 けていること,お よび初期土壌水分量の分布 が降雨浸

透 に与 える効果等について考察 した。

(農 土論 集169,pp.1～10)キー ワー ド 降雨浸透,浸 潤前線,動 水含水率,毛 管懸垂水,開 放不

飽和毛管浸透,動 的平衡浸透,毛 管水縁

〔研究論 文〕

FINITE ELEMENT METHOD ANALYSIS OFMICROIRRIGATION SYSTEM PRESSURE

DISTRIBUTION(本 文=英 文)

康 躍 虎 ・西 山壮一

潅水施設の設計の目的 は均等散水 にあ ると言っても過

言ではない。吐出流量が水頭の関数であ るため,均 等散

水の検討 に対 して,水 頭分布を求 める必要が生 じる。本

論文では,ラ テラル管 と給水管 とを一体 として,有 限要

素法によ り水理解析 を行い,そ の解析の有効性 を論 じた。

また,ド リップ潅概 システムを効率的に解析 する手法

を論 じた。

(農 土 論 集169,pp.19～26)

キー ワー ド 有限要素法,マ イクロ潅概,圧 力分布

〔研究論文〕水撃モデルによるパイプライン非定常流(高周波,低 周波)および定常流の統一的な

数値解析法- 混合型時間補間特性法と最適計算格子形成法-

島田正志

大規模パ イプラインの非定常流(高 周波,低 周波)お よび定常流を水撃モデル で統一的に効率的に解析 しうる混

合型時間補間特性法 を提案 した。本法で は,補 間 による

数値振幅減衰,分 散 の評価に より適切な時間ステップを決 め,最 適 な格子形成 を行 う。特 に,短 い管路が ある場

合 には,陰的時間補 間により,飛躍的 に大 きな時間 ステップを使用できる。パ イプライ ンモデル に対する数値計算

の結果か ら,混 合型時間補間特性法の有効性 を検証 したが,低 周波流れ解析 でも,圧 力振動 を再現 し,弁 遮断問

題 も容易に扱 えるな ど,剛 性 モデル解析にない長所があ

る。

(農 土 論 集169,pp.35～45)キー ワー ド パイプライン,水 撃解析,特 性直線法,陰 解法,陽 解法,

低周波,定 常流,安 定性解析,格 子形成

〔研究論文〕先行降雨のもとでの降雨浸透現象

- 降雨浸透現象の研究(III)-

富永雅樹

先行降雨があった場合の降雨浸透現象を明 らか にする

ために実験 と考察 を行った。深さ5mの 円柱状の模擬地

盤 に,30mm/h×2hの 降雨入力 を毎 日同 じ時刻か ら5

日間与 える実験 を行 った結果 では,2日 目以降の降雨入

力 によって,す でに降下の途 中にある1次 的な浸潤前線

の上部の浸透帯 に2次 的な浸潤前線が出現 した。2次 的

な浸潤前 線 は土壌水 分の増加 につれ降下速 度が速 くな

り,1次 的な浸潤前線の中に吸収 され,そ の降下運動を

継続 させ ることなどがわかった。 これ らの現象 は,先 行

降雨が開放不飽和毛管浸透 モデル にお ける毛管懸垂水を

発達 させ ると考 える ことによ り説明で きる ことを示 し

た。

(農 土論 集169,pp.11～18)

キ ー ワー ド 降雨浸透,浸 潤前線,毛 管懸垂水,土 壌水分量分布,開

放不飽和毛管浸透,先 行降雨,毛 管水縁

〔研究論文〕高炉セメン トを用いた暑中コンクリー トの

凝結時間ならびに圧縮強度特性

月岡 存

高温(30～33゜C)条 件 下において,高 炉 セメン ト中の

高炉ス ラグ微粉末の含有量 と使用混和剤の種類(標 準形

と遅延形)が,コ ンクリー トの凝結時間,お よび養生条

件 と材令の異 なるコンクリー トの圧縮強度 に及ぼす影響

について,実 験的 に検討 した。

その結果,(1)高 炉セメ ン トの使用 は普通 ボル トランド

セメン トより,ま た,遅 延形 の混和剤の使用 は,コ ンク

リー トの凝結時間の遅延 に効果が ある。(2)高炉セ メン ト

コンクリー トは使用混和 剤の種類 にかかわ らず,長 期材

令 にお ける強度増進が大である。(3)高温乾燥 の下では,

コンクリー トの材料や種類 よ りも脱型材令 が圧縮強度に

大 き く影響 する,こ とな どを明 らか にした。

(農 土論 集169,pp.27～33)

キ ー ワー ド 高炉セメン ト,暑 中 コンクリー ト,コ ンクリー トの凝結

時間,圧 縮強度,高 温養生

〔研究報文 〕

釧 路 東 地 区 に お け る地 盤 改 良 工 事

千葉 佳彦 ・金 山慎 一 ・上 野 賢治 ・日下部 史明

広域農道釧路東地区は北海道東部,釧 路支庁管内のほ

ぼ中央部に位置す る。 この うち,第 一期工事 区間 は,軟

弱地盤である釧路湿原内を通過す る。広里大橋部 は,す

べ り破壊,残 留沈下,橋 台の側方流動,導 水 カルバー ト

の変形な どが予測 された。 これ らの対策 として,地 盤 改

良 を行 うこととし,橋 台背面部 には深層混合処理工法,

道路部には砕石 コンパ クシ ョンパイル工法 を施工 した。

これ らの工事概要 を示す とともに,各 工法 の設計法,

施工手順,施 工管理方法な どについて報告 する。

(農 土論 集169,pp.47～55)キ ー ワー ド 地盤改良,砕 石コンパクションパイル工法,深 層混合処

理工法,広 域農道,農 業経営の安定