Nahid Paper 1 Final ver 2 1 - Sudan University of Science ... · 1 Optimal Pipelines Sizing for...

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Optimal Pipelines Sizing for Water Distribution Systems of Micro

Irrigation

Hassan Ibrahim Mohamed1, Nahid Ibrahim Ahmed El Haj

2, Haitham Ragab El Ramlawi

3,Abd

El Rahman Mohammed Nour4 , and Omran Musa Abbas

1

1-Department of agricultural engineering, College of agricultural studies Sudan University of science

and technology. 2- Department of Agricultural Engineering. Khartoum State- Ministry of Agriculture

and Water Resources 3-Centre of Dry land Farming Research and Studies, Faculty of Agricultural and

Environmental Sciences, University of Gadaref 4-Department of agricultural engineering, College of

agricultural Technology and Fish sciences -Neeleen University.

ABSTRACT

The main aim of the present study is to develop optimization process of water pipe networks

design through studying the effect of pipe and pump costs on the optimization process in a

network with a predetermined layout. Computer simulation and analytical solutions have

been used for minimizing the Capital cost of multiple-outlet pipelines of drip water

distribution system composed of many pipes with different diameters, while the variation in

pressure head is restricted to assure the required outlet discharge uniformity. The present

study focuses on developing a cost function design model for pump-pipelines system with

emitters distributed along the pipelines. The model has been developed to support the

design of micro irrigation systems and to advise farmers to improve drip system

performance .The procedure uses the linear programming, in which pipe length with

commercial pipe diameter is used as optimum variable and the factors affecting the pressure

head along multiple-outlet pipelines are considered. The objective function, to obtain the

optimal diameter of each pipe, is based on the capital cost of the piping system, pumps and

the cost of energy required for operating the system. The model consists of an objective

function that maximizes profit at the farm level, subject to appropriate geometric and

hydraulic constraints. The main items considered affecting cost were: operating pressure

within the network, the pipe diameter, and the pumping cost which include pumping station

and energy. The hydraulic and operational limitations imposed on the system are: limiting

maximum and minimum velocity, permissible pipe pressure, defined pressure at outlets, and

defined discharge at selected points.

Model solution employ WINQSB 6.0, and uses Excel database with information on emitters

and pipes available in the market, as well as on crops, soils and the systems under design.

The solution procedure is based on an iterative scheme and examined two case studies.

Results obtained indicate model capability to cut down costs of pipes as compared to

conventional design method.

Keywords: optimum network, water pipe network, water least annual cost principle, linear

programming, pump-pipelines system design, micro - irrigation network.

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1. INTRODUCTION

Sustainable irrigated agriculture requires irrigation practices that are environmentally

friendly, economically viable and lead to high irrigation performance (Pereira et al., 2002).

Micro irrigation systems have the potential for achieving high irrigation performance and

offer a large degree of control, enabling accurate water and fertilizer applications according

to crop water and nutrients requirements, thereby minimizing environmental impacts and

providing for increased performance and water productivity. Achieving this requires that

systems are designed and operated in a way that water is applied at a rate, duration and

frequency that maximize water and nutrient uptake by the crop, while minimizing the

leaching of nutrients and chemicals out of the root zone (Hanson et al., 2006). Highly

uniform and timely water application is therefore required (Mermoud et al., 2005; Santos,

1996; Hanson et al., 2006). Drip water distribution network is a system of hydraulic elements

contains (pipes, reservoirs, pumps, valves of different types), which are connected together

to provide the quantities of water within prescribed pressures from sources to the plant.

Hence, micro irrigation system need to be designed and operated based on the selection of

pipeline sizes which is directed to achieve high uniformity of water distribution and

minimum operating costs. The selection of a pipeline size to meet a specific criterion, such as

the minimum annual expenses, has been extensively treated by various researchers

(Haghighi et al., 1989; Barragan et al., 2006; Kang and Nishiyama, 1996; 2002; Demir et al.,

2007; Valiantzas, 2003; Valiantzas et al., 2007). The basic components of a micro -irrigation

system are: the pump/filtration station (consisting of the pump, filtration equipment,

controllers, main pressure regulators, control valves, water-measuring devices and chemical

injection equipment); the delivery system, (includes: the main and sub main pipelines to

transfer water from the source to the manifolds, (filters, pressure regulators, and control

valves); the manifolds, which in turn supply water to the laterals and the laterals that carry

water to the emitters (Pereira and Trout, 1999; Evans et al., 2007). Design of micro irrigation

systems is therefore complex considering the need to select and size all system components

and the need to design for a targeted uniformity of water application (Bralts et al., 1987;

Keller and Bliesner, 1990; Wu and Barragan, 2000).

Main advances in design of micro irrigation systems refer to pipe sizing and layout and to the

selection of emitters because these system components control the potential irrigation

performance and costs. The design options relative to the pump, valves, controllers, filters

and fertilizer devices are generally made after pipes and emitters are selected since they

depend upon related pressure and discharges at the various nodes of the system network

3

(Keller and Bliesner, 1990). However, their appropriate selection also influences the

irrigation performance, and they also produce additional head losses that must be

considered when sizing the system. To support and ease design, a variety of models have

been developed such as that for the pump/filtration station (Haghighi et al., 1989), for

assessing emitter uniformity (Barragan et al., 2006), for pipe sizing (Kang and Nishiyama,

1996; 2002; Demir et al., 2007) and for economic optimization of systems (Valiantzas, 2003;

Valiantzas et al., 2007).

Optimization methods for pipe network analysis has been reviewed by Stephenson et al,

(1981) .Their extensive study showed that the dynamic programming schemes are suitable

for pipe size selection of main pipelines. Transportation programming is convenient for cases

in which the pipe routes and size are to be optimally selected. Because of the complex

network analysis schemes algorithms based on linear programming technique has been

developed to minimize system costs (Valiantzas, 2003; Valiantzas et al., 2007). With the

advances in optimization search methods, network analysis has been extended to include

network routes (Cembrowicz et al,. 1996) and other important parameters such as:

management of irrigation systems (Srinivasan and Guimaraes, 1996, Eduardo and Marino,

1990 and Mohtar et al 1991) where the effects of land topography, irrigation method and

land allocation, maximization of land yield, profit and/or management of wastewater reuse

(Afshar and Miguel, 1989) is included. Application of the genetic algorithms for pipe network

optimization is in progress where it may provide some advantages over the classical linear,

dynamic and/or nonlinear programming methods (Dandy et al, 1996, Simpson et al ,1994).

The present work is directed to the study of a single source pump-branching pipes micro

irrigation distribution system for optimum selection of the pipes diameters on the basis of

minimum cost function under limits of hydraulic constraints for achieving high irrigation

performance using linear programming technique.

2. THE MATHEMATICAL MODEL The decision support model was developed to design drip and micro sprinkling systems, and

as a tool to advice farmers about how to improve their micro irrigation systems when using

data obtained during field evaluation of systems under operation. It is employs WINQSP, and

Excel programs and runs in a Windows environment in a personal computer.

The conceptual structure of the model is in presented figure 1, where two main components

are identified: the database and the Hydraulic Procedure. The algorithm is mainly oriented

to design and select the pipe system and emitters for an irrigation sector. The computer

design model has been integrated into three principal modules: the database and layout

module, the design module and the evaluation module. The database module entails

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specifying details of the system alignments and contains information on emitters, pipes, and

crops, soils such as: the emitter and lateral spacing; the lengths of the laterals and the

manifold; and the elevations of the pipes. Once this has been done, then the actual pipe

design can be carried out in the design module. The design module is tailored to iteratively

size the pipe and emitters system for individual laterals using WINQSB algorithm. Output

from this procedure includes: a display of the pipe flow and pressure characteristics;

allowed pressure envelope and the pipe hydraulic grade-line; and a summary table of the

operations performed in the design process.

2.1 Characteristic of the model: The model has been developed for fields with known dimensions on flat terrain. The water

source is assumed to be groundwater provided by a pump located at the centre of field. All

sub main pipes that feed the sub-units via supply pipes are perpendicular to the main lines

and are fed from both sides of the main lines. All pipes are made from polyethylene, and

emitters are fixed on the laterals at a fixed spacing. Each supply, sub main and mainline pipe

is controlled by one independent valve, which is located just at the beginning of the

corresponding pipe. One filter unit is assumed to be located just after the pump. Water is

assumed to be extracted from groundwater by means of a turbine pump system. The main

and sub main pipes are buried while sub-unit pipes (laterals, manifold, supply) are laid on

the ground. Total system cost consists of capital and installation costs plus the present value

of the operating costs over the expected life of the project.

2.2 Formulation of the model:

2.2.1 Model assumption and data input: In the present optimization model the general

configuration of pipes within the field (main and sub main lines) and within the sub-units

(lateral, manifold and supply lines) is fixed. However, since the area and the dimensions of

sub-units in the both X and Y directions change in each iteration of the field division, the

length and the size of all pipes change as well. The model was developed for a field with

given area and known dimensions for which the water source is located at the centre of

field.

2.2.2 Optimization procedure: The model evaluates all combinations of pipe sizes, and shift

patterns. The system cost is evaluated for various pipe sizes. Optimization is carried out by

complete enumeration of all alternatives. The following values are assumed to be known:

(1) The dimensions of the field,,[Fx ,m and Fy, m];

(2) The depth of the water table [Hwt ,m];

(3) The potential evapotranspiration,[ETo mm/day], the crop coefficient, [Kc];

(4) The minimum and maximum percentage of wetted area,[Pw, %];

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(5) The application efficiency of drip irrigation, [Ea, %];

(6) The annual irrigation requirement for the crop, [Air, mm];

(7) The field capacity, FC and the permanent wilting point[ PWP], of soil;

(8) The depth of root zone, [R, m], soil infiltration rate, Isoil (mm/hr) and soil bulk

density Row S (g/cm2);

(9) The portion of the available moisture depletion f (%);

(10) The spacing between emitters,[ dx ], and laterals, dy, respectively (m);

(11) The pipe cost coefficients[ k1,k2, k3]; the pump, cost parameters[ k, a, b];

(12) Efficiencies for the electric motor,[Em], and pump,[ Ep], respectively;

(13) The discount rate,[ i ], and expected project life,[ n, years];

2.2.3 The decision variables in the design of irrigation:

1. The pipe diameters and locations.

2. Pump locations and sizes.

3. Valve and regulator locations.

4. Reservoir sizes and locations.

The objective function: The objective of this optimization is to establish the quantities of the

each of the decision variables that will be used in the irrigation network so as to minimize a

pre-selected criterion. As mentioned previously, the most common criterion for the

optimization of network design is that of minimum cost. In order to achieve this

optimization, the value of the selected criterion is expressed as a function of the quantities

of the decision variables being used in the design.

The decision variables: Normally the quantities of the decision variables that can be used will

be constrained by a number of pre-determined factors. When considering pipe diameters as

decision variables for the design of irrigation networks, the allowable diameters to be used

along a specific path in the network must be equal to or greater than the smallest possible

diameter that will result in the pre-determined allowable head loss along the path. So for

every optimization problem, a set of relationships can be formulated which govern the

quantities of the decision variables that may be incorporated into the objective function.

These relationships are referred to as the constraints of the optimization.

Thus the optimization of the design of single-source branching networks can be formulated

as an LP problem, and it is derived in accordance with Karmeli et al, (1968) as follows:

a- The links of a network are defined by the upstream node, i, and the downstream node, j,

respectively. Then for each link ij in the network being designed a candidate set of

diameters, m, is to be considered. One set of decision variables is given by the length of pipe

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for each of the candidate diameters to be used in each link ij. These lengths are expressed as

X�� and they have associated costs given by C�� per unit length.

b- A second set of decision variables is given by the operating head of the pump at the

source of the network, for each different network loading condition, ℓ. These operating

heads are expressed as XP (l). The discharge at the pump for loading ℓ is QP (ℓ), and k (ℓ) is

the present value of the operating cost of the pump per unit of head and discharge that it

delivers, multiplied by: (a) coefficients reflecting the units in which the head and discharge

are expressed, (b) the efficiency of the pump and (c) the fraction of the total pumping time

during which loading ℓ is operative. Then the cost of operating the pump during loading ℓ is

given by k (ℓ).XP (ℓ) QP (ℓ). Likewise, if we assume that the capital cost of the pump, kc per

unit of power, increases linearly with its maximum operating head XPM, then the overall

capital cost of the pump at the source is given by kc XPM.

The drip irrigation design model described in this paper consists of small permanent system

with semi-automation, thus labor cost is considered to be small.

2.2.4 The Objective Function: The drip irrigation design model described in this paper

consists of an objective function that minimizes the sum of the capital cost and present

value of operating cost subject to appropriate constraints. The system is assumed to be

permanent with semi-automation, thus labor cost is considered to be small. In order to

formulate and solve an optimization problem mathematically it is first necessary to define

the decision variables. The objective function of the LP is given by:

Minimize [K] = ∑ij ∑m

C ijm

Xijm +

∑l

K (l)*XP (l)

*QP (l) +

Kc XPM

----------------------------- (1)

Where: ∑

l the summation of all loading, ∑

m the summation of all candidate pipe diameters m in link ij, ∑

ij the summation of all links ij in the network,

K= the total (present value) cost of the system.

2.2.5 The Constraints: The number of constraints in many cases a large network will generate

a very large LP, consisting of many variables and constraints. In order to maintain the

problem within practical bounds so that it can be solved on readily available computers, it is

particularly necessary to limit the number of constraints. This can generally be done by

careful selection of the head constrains. It will not be necessary to specify these constraints

for each and every node in the network. Hence, the LP should be run initially with a small set

of constraints covering what the designer considers to be the most critical nodes. In many

cases, the combination of topographic effects and the minimum or maximum head

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requirements for ‘less critical’ nodes will ensure that these requirements are automatically

satisfied by specifying the constraints for the ‘more critical’ nodes. The designer should

always check that all the head requirements are met in the optimal solution. If they are not,

then constraints should be added for the nodes that do not meet their respective maximum

or minimum head requirements, and the LP should then be re-run. Four sets of constraints

can be derived for the LP problem:

A- The non-negativity constraints: It is a condition of the LP solution procedure that the

decision variables must all be non-negativity. Thus a set of constrains can be stated as:

XIJM ≥ 0 ------------------------------------------------------------------------------------------- (2)

XP (l) ≥ 0 ……………………………………………………………………………… (3)

B- Length constraints: The lengths of each diameter of pipe selection for a given link must

add up to the length of the link. This is stated by:

∑m X ijm = Lij ………………………………...………………………………………… (4)

Where: Lij= the length of link ij.

C- The head loss constraint: Assume that the head at a specific reference node is fixed and

known for loading (l) and that either the maximum or the minimum (or both) allowable

heads at a second node downstream of the reference node are also known for loading (l).

Then the total head loss in the network along the path from the reference node to the

second node must not result in the head at the second node being either greater than the

allowable maximum, or less than the allowable minimum. This is expressed by:

HMINn(l) ≤ Hs(l) ±∑ij∑mJijm (l) Xi jmHMAXn…………………………………………….(5)

Where: Hs (l) = the head plus elevation at the reference node, s, for loading (l).

HMINn (l); HMAXn (l) = the minimum and maximum allowable head plus elevation at the

second node, (n), for loading (l).

Jijm(l) = the head loss (or gain depending on the direction of flow) per unit length of pipe of

diameter m in link (ij) under load (flow)( l).

∑ij ∑m = the summation of all segments (Xijm) in all links (ij) along the path from node s to

node (n).

Jijm (l) is calculated from the Hazen-Williams (or some other appropriate) formula.

D- The pumping constrain: Since the operating head of the pump for each loading is a

decision variable in the LP, it must be constrained not to exceed the maximum possible head

that the pump is capable of producing, XPM. This is stated by:

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XP (l) ≤ PM……………………...………………………………………………………..……… (6)

Thus when the LP defined by the objective function and the four sets of constraints

described above is solved for a specific network, the solution provides the designer with the

pipe diameters and pumping head at the source that should be used so as to give the

cheapest possible network that will satisfy the pre-determined hydraulic requirements. It

should be noted that the LP solution provides a consistent hydraulic solution for the network

at the same time as it optimizes the design.

2.3 Model Assumptions:

A-Pump costs: In formulating the objective function of the LP, two assumptions were made

about the capital cost of installing pumps in an irrigation system.

I-Firstly it was assumed that the cost increases linearly with increasing capacity (as

measured in units of power) of the pumps. This assumption is in fact an incorrect one,

since the capital cost per unit of power decreases with increasing capacity, due to the

various economies of scale. This would suggest the use of a technique known as

separable programming, which uses linear approximations for separate portions of the

cost against capacity curve. However, Alperovits and Shamir (1977) have proposed a

simple iterative algorithm based on the LP solution.

II-Secondly it was assumed that the dominant parameter determining the required

capacity is the maximum head required at the pump. The second assumption implies

that the discharge at the pumps will be constant for all loading conditions. While there

are several advantages in trying to achieve this for an irrigation network, it will

normally not be feasible. The discharge requirements at the pump will usually vary

during the irrigation cycle. The objective function will therefore only be valid when

XPM is associated with the maximum power requirement at the pumps, i.e. when:

XPM*QP (L) > = XP (t).Qp (l) ………………………………...……… (7)

Where: L = the loading condition associated with XPM.

B-Candidate diameters: In setting up the LP for a network, the designer has to specify a set

of candidate diameters, m, for each link ij. This results in introducing an implicit constraint

into the LP since the optimal solution cannot include any segments, (Xijm) of a diameter m

that was not in the initial candidate set. However, this may be overcome by using the

iterative procedure of Alperovits and Shamir (1977).

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12

3. DATA COLLECTION AND ANALYSIS

3.1-Location of the Study Area: The study is conducted in A/Gabar Farm which located in

the Northern part of State of Khartoum (Alkadaro), is located at eastern side of the River

Nile and at 13km north Khartoum (Bahri).It lies between latitudes 15º 37´ 40" and longitude

32º 31´51".

3.2-Drip System in Abd elgabar Farm: The first numerical case study is A/Gabar Farm which

consist of a2-branch system with 6 draw-off points along each branch (Fig 1).A/Gabar Farm

Input Data includes:

a- Well and Tank: The drip irrigation system under study was applied with water from

a well in the farm, through storage tank. Then pumped into a fish pond of capacity of

2890 m3.

b- Pump unit: A centrifugal pump operated by electric motor (7.5 kW), was used to

draw the irrigation water from the storage tank to supply the system. This setup gave

pressure of 3 bars in the main line.

c- Control unit: the control unit consist of the following:

1. Discharge valve to control the water follow in the system.

2. Pressure-reducing valve to control the pressure in the system.

3. Cleaning or flushing valve.

d- Filtration system: The water was conveyed from the fish pond by 75 mm (3")

diameter, pipe and then paned through two sand filters.

e- Fertilizer unit: Fertilizer and chemical injectors were incorporated with the drip

system to supply fertilizers. Herbicides, insecticides, fungicides, trace elements,

nutrient solution and acid at frequent or nearly continuous application with the

irrigation water.

f- The main line: the main pipe line was mad of polyvinyl chloride (PVC). It was 240 m

long and 75 mm (3") in diameter.

g- Sub main line: the sub-main pipe was also made of polyvinyl chloride (PVC) the sub-

main line was 240 m and 63 mm( 2.5") in diameter

h- The lateral lines: the lateral pipes were made of black linear low density

polyethylene (LLDPE). There were 80 laterals, each 40 m long and 16 mm inside

diameter. Laterals were joined to the sub main at 1 m spacing.

i- Emitters (drippers): Emitter with 40 cm spacing along the laterals, each with

average discharge of 1.08 I/h.

13

Figure 2: Schematic diagram for case one: A/Gabar Farm.

The following data was collected from the farm:

(1) The dimensions of the field, F� (m) and F� (m);

(2) The depth of the water table H� (m);

(3) The potential evapotranspiration, ET� (mm/day), the crop coefficient, K�;

(4) The minimum and maximum percentage of wetted area, P, (%);

Water

Source

Pum

p Filte

rs

Magneti

c

23 m 87 m

2.3

m

40 m

3 in

1.5 in

3 in

2.5

in

14

(5) The application efficiency of drip irrigation, E� (%);

(6) The annual irrigation requirement for the crop, A��, (mm);

(7) The field capacity, FC and the permanent wilting point PWP, of soil;

(8) The depth of root zone, R, (m), soil infiltration rate, I�� (mm/hr) and soil bulk

density γ� (g/cm�);

(9) The portion of the available moisture depletion f (%);

(10) The spacing between emitters, d�, and laterals, d�, respectively (m);

(11) The pipe cost coefficients k�,k ,k� �; the pump, cost parameters k, a, b;

(12) Efficiencies for the electric motor, η�, and pump, η", respectively;

(13) The discount rate, i , and expected project life, n, (years);

3.3-Climate: The study site is within the semi-desert it receives rain fall about(150-

170mm/year),The seasonality and variability both characterize the rainfall in time and space

.Generally in January the average temperature rises from (14cº) at dawn to (30cº) in after

noon, while in May the hottest month it rises from (25cº to 42cº) rainfall start in(June - July),

and may continue up to September in dry years, Winter is know with it is prevailing strong

northerly wind, which causes serious sand and dusty storms. Meteorological data reported

by Khartoum North Meteorological Station at Shambat ,concerning precipitation, P (mm)

and weather data (maximum and minimum temperature, rainfall, wind speed, bright

sunshine hours and relative humidity were taken for the period of 30 years (1961 – 2006)) is

used to compute reference evapotranspiration, ETo (mm), using FAO Penman-Monteith

methodology described by James (1988).

3.4-Soil: The soil is sandy -clay to clay. Soil moisture is one of the most limiting factors in the

area. The soil of study area is predominantly Aridisols with pockets of Vertisols formed on

old alluvium deposits and Entisols on recent alluvium and Aeolian deposits .Most of the soils

of the site are salt-affected (Ali et al 2009).

The soil of A/Gabar farm can be described according to United State Department of

Agriculture Soil Classification Chart as sandy clay loam soil. The soil is alkaline with 7.8 pH,

and characterized with low water holding capacity (36% moisture at field capacity, 26 %

permanent wilting point on volume basis and 1.3 gm/cm^3 bulk density. The measured and

collected input data is as follows:

3.5-Soil mechanical analysis: Two locations were randomly selected to represent the soil

under study. Three soil samples were taken from each location at depths 0-30, 30-60, 60-90

cm. Soil texture was determined using the hydrometer method as described by Blake (1965).

Chemical analysis: Two samples of water were taken to study their chemical properties.

15

3.6-Infiltration characteristics: Three representative sites were selected for measuring

infiltration rate using the double ring infiltrometer as described by James (1988).The average

initial infiltration rate is 12.3 cm/h and the final rate is 1.8 cm/h. The relation between intake

rate and elapse time can be approximated by the power relation of the form:

Y = 12.48X ^ 0.84 ……………………………………………...…………….. (8)

Where: y = infiltration rate and X = time in minutes. According to James (1988) the high final

intake rate values exhibited by the soil indicates that the farm is more suitable to be

irrigated by drip irrigation than by other surface methods.

3.7-Determination of crop water requirement (CWR): Crop water requirements are usually

expressed in units of water volume per unit land area (m3/ha), depth per unit time

(mm/day). Crop water requirement, which is equal to crop evapotranspiration, is estimation

according to the following equation (Jensen, 1983):

Etc = ETo x Kc x Kr -------------------------------------------------------------------------------- (9)

Where:

ETc = Crop evapotranspiration (mm / day).

ETo = Reference crop evapotranspriration (mm/day).

Kc = Crop coefficient.

Kr = Reduction coefficient

The crop water requirement for each month is calculated using the following equation:

CWR = Etc x days of the month………………………………………………… (10)

Where: ETc = Crop water requirement (mm/day).

The net crop water requirement (NCWR): The net crop water requirement was calculated by

subtracting the monthly effective rainfall (ERF) as:

NCWR = CWR – ERF …………………………………….……………………………….……… (11)

The effective monthly rainfall (EFR, mm) was calculated from the total rainfall (TRF, mm)

according to the USDA soil conservation service empirical relationships.

3.8-The uniformity of the drip system: Field data was collected from the drip system

installed at A/Gabar Farm (Figure 2). The discharge from 70 emitters (randomly selected)

was used to test the uniformity of the system. The uniformity of the system (Eu%) was then

calculated using the following formula:

Eu% = 100#$

#%&' -------------------------------------------------------------------------------------- (12)

3.9-Crop water use efficiency: The water utilization by the crop is generally described by the

following formula (Karmeli et al, 1968):

WUE (kg/halm3) = Crop yield (kg/ha)/ water supply (m3) ---------------------------------- (13)

16

Measurement of water discharge in the drip system: Measurements were taken from 14

laterals using catch cans, a measuring cylinder and a stop watch. The pressure was adjusted

at 1bar over all laterals, and was replicated at least three times for each emitter. This

application was done every 0, 10, 20, 30, 40 meters along the lateral lines, one hour after

the system operation.

3.10-Measurement of water discharge in the drip system of A/Gabar Farm.: Measurements

were taken from 14 laterals using catch cans, a measuring cylinder and a stop watch. The

pressure was adjusted at 1bar over all laterals, and was replicated at least three times for

each emitter. This application was done every 0, 10, 20, 30, 40 meters along the lateral lines,

one hour after the system operation. The collected data was then expressed by a linear

regression as described by Gomez and Gomez (1984).

3.11-Summary of Farm Input Data: The input data collected from A/Gabar farm is

summarized in table 1, 2.3, 4 and 5.

Table 1: Summary of Farm Input Data

No. Particulars Unit Farm Data

1. Nominal Discharge of Emitter lit/hr 4.00

2. Soil Saturated Hydraulic Conductivity mm/hr 25

3. Initial Soil Moisture Content % 14.00

4. Final Soil Moisture Content % 35.00

5. Wetted Depth m 1.50

6. Chose Soil Type 5.70

7. Distance Between Trees m 0.4x1

8. Max. Daily Crop Water Req. mm/day 5

9. Diameter Of Shading m 0.4

10. Irrigation Efficiency % 85

11. Percent Of Ground Cover % 85

12. Coefficient Of Emitter Equation -- 0.855

13. Exponent Of Emitter Equation -- 0.67

14. Length Of Last Lateral m 40

15. Length Of Last Manifold m 40

16. Length Of Last Sub Main m 327

17. Allowable Irrigation Time hr 4

18. No. of Plants Rows/Manifold No 40

19. No. of Plants/Lateral No 100

20. Distance of First Emitter Full Space

21. Lateral Layout One Side

22. Manifold Layout One Side

23. Available Size Of Lateral (Internal Diameter) mm 13.6

24. Distance Of First Lateral Full Space

25. Available Size Of Manifold (Internal Diameter) mm 46.4

26. Available Size Of Sub Main (Internal Diameter) mm 84.6

27. Number of Manifold / Sub Main Mo 6

28. Number of Sub Main / Mainline No 1

29. Available Size Of Mainline (Internal Diameter) mm 84.6

30. Length Of Mainline m 23

31. Static Suction Head m 0

32. Net Positive Suction Head m 4

17

33. Head Losses in Control Head m 15

34. Pump Efficiency % 35

35. Engine Efficiency % 65

Table 2: Input Data of A/Gabar Farm for LP optimization Model

Link 12 D Q K* 108

C Q(lit/sec) J (m/m) FL (m)

Length (m) 121 101.6 28.236 121 150 7.8 0.00854 0.19642

23 122 76.2 28.236 121 150 7.8 0.03465 0.79695

123 63.5 28.236 1121 150 7.8 0.0842 1.9366

Link 23 D Q K C Q(lit/sec) J (m/m) FL (m)

Length (m) 231 101.6 23.53 121 150 6.5 0.00609 0.2436

40 232 76.2 23.53 121 150 6.5 0.02472 0.9888

233 63.5 23.53 121 150 6.5 0.06007 2.4028


Length (m) 341 101.6 18.824 121 150 5.2 0.00403 0.1612

40 342 76.2 18.824 121 150 5.2 0.01635 0.654

343 63.5 18.824 121 150 5.2 0.03974 1.5896


Length (m) 451 101.6 14.118 121 150 3.9 0.00236 0.0944

40 452 76.2 14.118 121 150 3.9 0.0096 0.384

453 63.5 14.118 121 150 3.9 0.02332 0.9328


Length (m) 561 76.2 9.412 121 150 2.6 0.00453 0.1812

40 562 63.5 9.412 121 150 2.6 0.01101 0.4404

563 50.8 9.412 121 150 2.6 0.03263 1.3052


Length (m) 671 76.2 4.706 121 150 1.3 0.00125 0.05

40 672 63.5 4.706 121 150 1.3 0.00305 0.122

673 50.8 4.706 121 150 1.3 0.00904 0.3616


Length (m) 281 101.6 28.236 121 150 7.8 0.00854 0.74298

87 282 76.2 28.236 121 150 7.8 0.03465 3.01455

283 63.5 28.236 121 150 7.8 0.0842 7.3254


Length (m) 891 101.6 23.53 121 150 6.5 0.00609 0.2436

40 892 76.2 23.53 121 150 6.5 0.02472 0.9888

893 63.5 23.53 121 150 6.5 0.06007 2.4028


Length (m) 9101 101.6 18.824 121 150 5.2 0.00403 0.1612

40 9102 76.2 18.824 121 150 5.2 0.01635 0.654

9103 63.5 18.824 121 150 5.2 0.03974 1.5896


Length (m) 10111 101.6 14.118 121 150 3.9 0.00236 0.0944

40 10112 76.2 14.118 121 150 3.9 0.0096 0.384

10113 63.5 14.118 121 150 3.9 0.02332 0.9328


Length (m) 11121 76.2 9.412 121 150 2.6 0.00453 0.1812

40 11122 63.5 9.412 121 150 2.6 0.01101 0.4404

11123 50.8 9.412 121 150 2.6 0.03263 1.3052

18


Length (m) 12131 76.2 4.706 121 150 1.3 0.00125 0.05

40 12132 63.5 4.706 121 150 1.3 0.00305 0.122

12133 50.8 4.706 121 150 1.3 0.00904 0.3616

Table 3: The elevation of irrigation system (A/Gabar Farm

Point Elevation

Proposed

Elevation m383

1 0.253 383.253 m

2 1.52 384.52 m

3 1.46 384.46 m

4 1.49 384.49 m

5 1.51 384.51 m

6 1.52 384.52 m

7 1.6 384.6 m

8 1.58 384.58 m

9 1.53 384.53 m

10 1.57 384.57 m

11 1.6 384.6 m

12 1.61 384.61 m

13 1.63 384.63 m

Price Int. Diam. Ex. Diam. (mm)

Ex. Diam.

(Inch)

7 101.6 4

5 76.2 3

4 63.5 2.5

3 50.8 2

2.5 25.4 1

Table 4: The head losses of drip irrigation system (A/Gabar Farm)

Item Value Unit

Average Operating Head of Emitter: 0.600 m

Total Head Losses in Lateral 0.030 m

Friction Losses in Manifold 0.199 m

Head Losses in Submain 6.960 m

Head Losses in Main 0.488 m

Minor Losses in Submain 1.390 m

Minor Losses in Main 0.098 m

Static Suction Head 0.000 m

NPSHR 4.000 m

Head Losses in Control 15.000 m

Difference in elevation between Pump and Highest Point in Main Line 1.327 m

Total Dynamic Head of Pump 28.705 m

Table 5: The head required at nodes (A/Gabar Farm)

Item Value Unit

Head required at node 1 28.705 m


19












3.11- Hypothetical case study- Karmeli, Peri and Todes Farm (Irrigation system design and

operation, 1985):

The schematic diagram for case two- Karmeli Farm is depicted in figure3.Data collected from

the farm includes: pipe length at each link, alternative diameters to use and their respective

costs-and shown in table6. The values of friction factors (J) are calculated for each diameter

of pipe from the Hazen Williams equation.

Figure 3: Schematic diagram for case two- Karmeli Farm

Table 6: Length of pipes at each link and pipe cost for Karmeli Hypothetical Farm

Link Length Candidate Diameter Pipe Cost

ij (m) 3 2 1 Diameter (mm)

Cost ($/m)

12 1000 500 450 400 100 16

23 1000 250 200 150 200 23

24 1000 450 400 350 250 32

45 1000 300 250 200 300 50

46 1000 450 400 350 350 60

67 1000 300 250 200 400 90

450 130

500 170

The values of friction factors (J) are calculated for each diameter of pipe using Hazen

Williams's equation. For example, for link 12, the values are as follows:

Candidate Diameter

Number

Diameter

(mm)

Q

(m3/hr)

J

(m/m)

NODE 1

ELEVATION.E1= 150 m

Water

Source

1 Pump

P =60 m

Q = 1120 m 3/hr

DISHARGE.d2 = 100 m3/hr d3 =100 m

3/ hr

E3 = 160

d4 = 120 m3/hr d5= 270 m3/hr

E5 = 150 m

d7 = 200 m3/hr d6 = 330 m

3/hr

E6 = 165 m E7 = 160 m

2 3

4 5

6 7

20

1 500 1120 0.00406 (J121)

2 450 1120 0.00678 (J122)

3 400 1120 0.01202 (J123)

3.10-Hydraulic Procedure for determining Pipe Diameters: The pipe diameters are

established using the allowed pressure variation in the system as a design parameter. The

general procedure for a given pipe involves first of all establishing a pressure envelope,

defined by the topographic elevations along the length of the pipe and the allowable

pressure variation within the pipe. The upper and lower limits of the envelope represent the

maximum and minimum allowable hydraulic grade lines respectively, along the pipe being

designed. Then, starting at the furthest end of the pipe with the smallest available diameter,

the pressure head in the pipe is calculated for points along its length, working back towards

its inlet. This pressure head will increase exponentially as the flow in the pipe increases

because more and more outlets are included along the length being considered. Considering

this curve in the other direction {i.e. in the direction of flow in the pipe), the exponential

shape represents the decreasing rate of head loss due to friction, per unit length, as the flow

in the pipe decreases. As soon as the actual hydraulic grade line for the diameter of pipe

being considered starts to rise steeply towards the upper limit of the allowed envelope, the

pipe is replaced by a larger diameter, thereby reducing the rate of pressure loss due to

friction. The process then continues until the inlet is reached. In this way a set of diameters

and their respective lengths are determined, such that the pressure variation in the lateral is

contained within the allowable limits (Karmeli et al, 1968).

3.12-Data Analysis: Statistical package for social sciences (SPSS) is used for analysis of

variance, regression, t-test, and Chi-squire test.

4. RESULTS AND DISCUSSIONS Model Application: Two case studies namely: Abd elgabar Farm (A/Gabar) and - Karmeli

Hypothetical Farm were investigated for purpose of model applications.

4.1- Abd elgabar Farm:

For purpose of model application to Abd elgabar Farm the LP problem is formulated as

shown in table 7.

Table7: Formulation of the LP problem for Abd elgabar Farm

21

Analysis of the input data as generated by QSB computer model is depicted in table 8.

Table 8: Results of analysis of linear optimization generated by using QSB - computer

program (A/Gabar Farm)

No. Constraint Left Hand Side Direction Right Hand Side Slack or Surplus Shadow Price

1 C1 390 >= 0 390 0

2 C2 610 >= 0 610 0

3 C3 0 >= 0 0 0

4 C4 0 >= 0 0 0

5 C5 331 >= 0 331 0

6 C6 669 >= 0 669 0

7 C7 1,000.00 >= 0 1,000.00 0

8 C8 0 >= 0 0 0

9 C9 0 >= 0 0 0

10 C10 0 >= 0 0 0

11 C11 518 >= 0 518 0

12 C12 482 >= 0 482 0

13 C13 0 >= 0 0 -57.1003

14 C14 1,000.00 >= 0 1,000.00 0

15 C15 0 >= 0 0 0

16 C16 126 >= 0 126 0

17 C17 874 >= 0 874 0

18 C18 0 >= 0 0 -20.8388

19 C19 1,000.00 = 1,000.00 0 127.3526

20 C20 1,000.00 = 1,000.00 0 16

21 C21 1,000.00 = 1,000.00 0 109.7519

22 C22 1,000.00 = 1,000.00 0 23

23 C23 1,000.00 = 1,000.00 0 77.9033

24 C24 1,000.00 = 1,000.00 0 47.1822

25 C25 53.5252 >= 30 23.5252 0

26 C26 40.0038 >= 40 0.0038 0

27 C27 48.3222 >= 35 13.3222 0

28 C28 30.0098 >= 30 0.0098 0

29 C29 45.0002 >= 45 0.0002 0

30 C30 40 >= 40 0 2,811.00

31 C31 59.9979 <= 60 0.0021 0

22

Application of linear programming optimization using QSB - computer program for A/ Gabar

farm is made using input data given in tables 1 to 5. The result is given in table 9 and

indicates that costs of all type of pipes are reduced significantly. This resulted in significant

reduction of total cost from 2390 to 1905 SDG.

Table 9: Significant Reduction of Total Cost (A/ Gabar Farm)

Parameter Cost of A/ Gabar Farm

Significance

Level Actual Model

Main Line Pipe 550 465.045 Sig

Sub main Pipe 1200 960 Sig

Lateral Pipe 640 480 Sig

Total Cost 2390 1905 Sig

4.2- Case Two: Karmeli Farm

When applying the model for the hypothetical farm of Karmeli the LP objective and

constraint are formulated as:

Minimize [K] = ∑ijCijmXijm for all ij e.g. ∑12 Cm X12 = 170 X121 + 130X122 + 90X123.

The constraints: 1- non-negativity: Xijm≥ 0 for all ij,m. 2- Length ∑mXijm = 1000 0 for all ij.

Head loss:

At node 2: 180 ≤ 210 - ( )� *+� **

At node 3: 190 ≤ 210 - ( ) �*+ �** − ( )� *+� **

At node 4: 185 ≤ 210 - ( ) .*+ .** − ( )� *+� **

At node 5: 180 ≤ 210 - ( )./*+./** − ( ) .*+ .** − ( )� *+� **

At node 6: 195 ≤ 210 - ( )./*+./** − ( ) .*+ .** − ( )� *+� **

At node 7: 190 ≤ 210 - ( J12�X12�� − ( J./�X./�� − ( J .�X .�� − ( J� �X� ��

Formulation the LP problem in typical QSB –model format is shown in table 10.

Table 10: Karmeli Farm LP Model Formulation

23

The results of analysis of linear optimization generated by using QSB - computer program for

Karmeli Farm is shown in table 11. The result indicates that the constraint for node 2 can be

left out because the constraint for node 3 will ensure that the node 2 constraint holds.

Similarly the constraint at node 6 ensures that the constraint at node 4 holds.

In the solution given above both links 12 and 24 have full lengths (1000 m) of only one

diameter. For link 12 this is the 450 mm diameter which was the middle candidate (m = 2)

for the link. Thus for this link the implicit constraint is not binding. However, for link 24 the

diameter chosen (450 mm) is for the candidate m = 1. In this case it is possible that the LP

wanted to look for possible diameters larger than 450 mm to get to a better optimum. In

order to eliminate the constraint we must re-run the LP with the candidate diameters for

link 24 changed to 500, 450 and 400 mm respectively.

Table 11: Results of analysis of linear optimization generated by using QSB - computer

program for Karmeli Farm

Combined Report for Karmeli 10% minor losses

Decision Solution Unit Cost or Total Reduced Basis

Variable Value Profit c(j) Contribution Cost Status

1 X121 390 170 66,300.00 56.3651 at bound

2 X122 610 130 79,300.00 23.7046 at bound

3 X123 0 90 0 0 basic

4 X231 0 32 0 16 at bound

24

5 X232 331 23 7,613.00 7 at bound

6 X233 669 16 10,704.00 0 basic

7 X341 1,000.00 130 130,000.00 34.8737 at bound

8 X342 0 90 0 6.2218 at bound

9 X343 0 60 0 0 basic

10 X1451 0 50 0 27 at bound

11 X452 518 32 16,576.00 9 at bound

12 X453 482 23 11,086.00 0 basic

13 X461 0 130 0 0 basic

14 X462 1,000.00 90 90,000.00 21.4349 at bound

15 X463 0 60 0 0 basic

16 X671 126 50 6,300.00 9.0582 at bound

17 X672 874 32 27,968.00 0 basic

18 X673 0 23 0 0 basic

19 MPX 59.9979 2,811.00 168,654.10 0 basic

20 X20 1 705.6 705.6 705.6 at bound

Objective Function (Min.) = 615,206.80

The overall results of analysis of linear optimization generated by using QSB - computer

program is given in table 12 .The results indicates that costs of all type of pipes is reduced

significantly and this resulted in significant reduction of total cost from 1235 to 649 SDG

given in Table 4.7. The results in Table 4.7 show that the increase in the cost of electricity is

associated with increase in pipe diameters but the relation is not linear. This case illustrates

the applicability of the analysis and software to handle a system with a large number of

draw-off points.

Table 12: Significant Reduction of Total Cost (Karmeli Farm)

Parameter Costs of Karmali farm Significant

Level Without optimization With LP optimization

Main Line Pipe 390 130 Sig

Sub main Pipe 560 370 Sig

Lateral Pipe 285 149 Sig

Total Cost 1235 649 Sig

5. CONCLUSIONS

The proposed model is based on the cost of piping system, pump and the energy consumed

to operate the system. The least annual cost principle is employed to determine the optimal

diameters of all piping segments.

Application of LP optimization model revels that there is significant reduction in the sizes of

main, sub main lines and lateral in all tested cases. This resulted in significant reduction in

total costs and improvement in the final designs.

25

The model is employed for a simple two parallel pipelines with 4000 emitters. A second case

study is also presented which considers the micro irrigated farm of Keramli et al (19187).It is

thus, highly recommended to apply the LP optimization module after hydraulic design for

purpose of reducing cost of pipes.

Practicability of the solution: Once the various iterative procedures and adjustments

described by the model have been carried out, and a satisfactory optimal solution has been

obtained, it should be checked for its practicability. For example, the solution may contain

segments for which the optimal length is too small to be practically significant, and these

should be eliminated in the final design specifications. The tendency in the past has been to

design irrigation networks with only one diameter pipe in each link. This will not normally be

the optimal design. In some cases, a particular link may be extremely long and the optimal

solution from the LP will be limited to specify only two segments of a different diameter

along its length. In this case it may be advantageous to specify a ‘dummy’ node at some

suitable point along the length of the link (e.g. at a sudden change of topography or at the

mid-point of the link).

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Cembrowicz, R. G., Ates, S. and Nguyen K. M.,1996 Mathematical Optimization of Main

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