Application of Numerical Newton-Raphson Method in Calculation of Emitter Water Discharge of Drip Irrigation System in “Mutis Cemerlang” Coffee Plantation

ABSTRACT


A. INTRODUCTION
Conditions of agricultural land such as the shape of the area, slope and elevation affect the quality of the soil which will greatly determine the level of quality of agricultural production (Khalil et al., 2015), (Nabiollahi et al., 2018), (Magdić et al., 2022), (Javed et al., 2022). Geographically, North Central Timor Regency comprises 2,669.70 Km 2 or approximately 5.48% of East Nusa Tenggara Province's land area. The topography of North Central Timor Regency is generally undulating and hilly with varying heights: 1) 177.60 Km 2 (6.63%) with an altitude of fewer than 100 meters above sea level (masl); 2) 1,499.45 Km 2 (56.17%) with an altitude of 100-500 masl; and 3) 993.19 Km 2 (37.20%) has a height of more than 500 masl. According to data from Agroclimate and Hydrology Research Institute in 2003, out of a total of 4.6 million ha in the East Nusa Tenggara region, approximately 1 million ha has an arid climate (semiarid) with annual rainfall <1000 mm, 7-10 dry months (<100 mm), and <2 wet months (> 200 mm).
Noepesu Village is one of 160 villages in the North Central Timor Regency with altitudes between 600 and 1200 meters above sea level and an average temperature of 18° C. With an average altitude of 1,000 meters above sea level and an average temperature of 18° C, Noepesu village has a moderate climate. Based on the division of climate zones and the types of plants that can grow in Indonesia, as stated by Frans Wilhelm Junghun, coffee can be successfully cultivated in Noepesu Village.
The entire area of Noepesu village is dry land with a hilly topography; most of the land has a ±45º slope and is cultivated as fields that are only planted once a year during the rainy season with corn, rice, cassava, string beans, and peanuts, among others. Typically, the rainy season lasts only a few weeks (4 months). Due to dry land conditions and associated constraints, a community's farming activities are restricted to land preparation before the rainy season and the rainy season itself. During the dry season, almost no farming occurs, with the exception of those who raise livestock.
Based on slope class, shape of the area and elevation, dry land in Nusa Tenggara can be grouped into three clusters (Mokarram & Hojati, 2016). Cluster A (<slopes 15%, <200 masl) is suitable for food crops, cluster B (slopes 15-40%, 200-700 masl) is ideal for food crops, forage for livestock, and plantations, while cluster C (slope > 40%, > 700 masl) is suitable for agroforestry. The selection of commodities and soil conservation efforts based on clusters is highly recommended for sustainability, and environmental sustainability, particularly clusters B and C. Highly recommended for cluster C region like Noepesu Village is a combination of annual and food crops; the steeper the land, the more annual crops should dominate (Derlukiewicz et al., 2020).
According to the 2014 numerical database book North Central Timor Regency, there are 26 farmer groups in the West Miomaffo District, including the Mutis Cemerlang farmer group. This group of farmers has 42 members and owns 11 hectares of farmland. This land is used to cultivate various crops, including garlic, red beans, carrots, potatoes, candlenuts, and coffee. The agricultural land of the farmer group has never been irrigated and relies solely on rainwater infiltration into the soil, so crop yields are highly dependent on rainfall. If rainfall is adequate, crop yields are satisfactory, and vice versa. Based on this water constraint, agricultural technologies such as drip irrigation supplying water continuously with low water discharge and high frequency (nearly continuously) around plant roots are required (Assis et al., 2014).
Numerous farmer groups have implemented drip irrigation as an agricultural technology in Noepesu Village, but the installation process did not account for pipe specifications (pipe length and diameter) and the condition of the agricultural land. This results in a short service life for drip irrigation. If this continues, farmer groups will be required to incur additional costs for installing a new system, resulting in higher installation costs. To optimize service life, drip irrigation pipe network systems require a hydraulics analysis method that considers pipe specifications and agricultural land conditions (Karim & Sahib, 2019).
The emitter is one of the most vital components of a drip irrigation system. The emitter or dropper is a component that continuously distributes water from the lateral pipe to the soil surrounding the plant with a low discharge and pressure close to the atmosphere (Hussain & Gupta, 2017). There is a component in the emitter that functions as a regulator for the amount of water discharged.
The hydraulics analysis of the drip irrigation network system is an analysis to determine the value of the water flow rate at the emitter (Abduraimova et al., 2022), (Ates, 2016). The flow rate at the emitter is represented by a nonlinear equation known as the closed pipe equation (Moosavian & Jaefarzadeh, 2014). Several numerical methods can be used to solve these equations, including Newton-Raphson, Hardy Croos (Brkić & Praks, 2019), Gradient (Teixeira et al., 2021) and the Linear Theory (Moosavian, 2017). The Newton-Raphson method has several advantages over other methods, including simultaneous network analysis, which makes the iteration process shorter and faster, and a low probability of discharge error correction (Saad et al., 2017). Based on pipe specifications, this study aims to apply the Newton-Raphson method to determine the optimal amount of water discharge at each emitter of a drip irrigation system. This study's findings can be used as a guide for utilizing sloping and arid land as agricultural land with the aid of drip irrigation.

B. METHODS
This quantitative study employs the Newton-Raphson method to determine the amount of water discharged by each emitter in a drip irrigation network system. From July to November 2021, this research was conducted at the Mutis Cemerlang Farmers Group coffee plantation in Noepesu Village, West Miomaffo District, North Central Timor Regency, East Nusa Tenggara Province. In addition to the direct collection of research-related data, several relevant journals and books were also consulted for additional references. This study employs the Newton-Raphson method and Matlab R2021 software for data processing.

Research Stages
The research stages are as follows: a. Looking for sources regarding the use of dry land for agriculture, and the application of the Newton-Raphson method in agriculture. b. Preparing land and pipe installation for drip irrigation. c. Numbering of all nodes, pipes and loop points. d. Construct nodal equations and pressure drop equations in the loop based on the drip irrigation network system that has been built. e. Constructing Jacobian matrices from the equations formed. f. Applying Newton-Raphson method iterations to obtain optimal solutions with the help of Matlab R2021 software.

Data Processing Stages
Pipeline analysis using the Newton-Raphson method is different from other analytical methods, since this method analyzes the entire network simultaneously (Ifiemi et al., 2020). The Newton-Raphson method is a reliable numerical method for solving systems of nonlinear equations. Suppose there are three nonlinear equations such as Equation (1) is expanded into a Taylor series, as follows: Equation (3) is then presented in matrix form, as follows: The solution to equation (4) is: Thus, the flow rate of Q1, Q2, and Q3 can be calculated as follows: The procedure for network analysis using the Newton-Raphson method begins in several stages, namely: a. Numbering of all nodes, pipes and loop points. b. Presenting the nodal equation as follows: where jn Q is the discharge in nth at node j, j q is the nodal withdrawal, and n j is the total number of pipes at node j. c. Presenting the pressure drop equation as follows: Where n k is total pipes in kth loop, and n K is the algebraic sum of the head loss in kth loop which can be expressed as i is pipe link number to be summed up in the loop k. d. Assume the initial discharge f. Determine the value of the partial derivative and function using the initial discharge pipe and . g. Determine ∆ with the form = Matrix. h. Using the obtained of ∆ values, the pipe discharges are modified and the process is repeated again untul the calculated ∆ valueds are very small.

C. RESULT AND DISCUSSION
The results and discussion are divided into two parts, land preparation and installation of drip irrigation networks, and calculation of emitter water discharge.

Land Preparation and Installation of Drip Irrigation Networks
The plantation land of the Mutis Cemerlang Farmer Group is agricultural land with a ±45 o slope cultivated as fields that are only planted once a year during the rainy season with corn, rice, cassava, string beans, and peanuts. The following illustrates the state of the land, as shown in Figure 1. Furthermore, the sloped land is transformed into terraces. This is done to shorten the slope and restrict or minimize runoff so water can percolate into the ground (Pramudo et al., 2016). The process of land preparation lasted three weeks. Researchers assisted members of the Mutis Cemerlang Farmers Group in the process of preparing the land. There are five terraces, each measuring 45 meters in length and varying in width from 3 to 5 meters in accordance with the terrain. This terrace serves as a stretch of HDPE pipelines with emitters for each planting hole containing coffee and tamarind plants. In addition, terraces are constructed so that water flow is distributed uniformly to all emitters in the row. The five constructed terraces are of varying heights, so the pipe requirements can be tailored to the existing conditions. The drip irrigation pipeline network installation phases occurred on 31 October 2021 and 7 November 2021, respectively. Researchers were assisted in the installation of the pipes by members of the Mutis Cemerlang Farmers Group. The drip irrigation system is composed of several parts (Jarwar et al., 2019): a. An emitter or dropper is a component that distributes water from the lateral pipe to the soil around the plants with a low discharge and a pressure close to atmospheric pressure. Discharge control components support the emitter used. b. Lateral pipe, is where the emitter is installed. The pipe used as the lateral pipe is HDPE pipe with a diameter of 0.5 inch (22 mm). The reason for using HDPE pipes as lateral pipes is that HDPE pipes are flexible so they can be adapted to land surface conditions and do not require pipe joints. c. The main manifold or sub pipe, is a pipe that distributes water to the lateral pipe. This pipe is designed using a 2-inch (60 mm) diameter PVC pipe. d. A component that distributes water to the manifold pipe is the main pipe. This pipe is constructed from 1 inch-diameter PVC pipe (32 mm). PVC (Polyvinyl Chloride) pipes are used as manifold and main pipes because they are commonly referred to as paralon pipes. The formation is made entirely of plastic or a combination of plastic and vinyl so that the pipe is stronger, lighter, more durable, and will not rust. e. Other supporting components consist of reservoirs, faucet stops, and valves.
The designed irrigation network system consists of 250 emitters arranged in 13 rows of combination for tamarind and coffee tillers. Each row of each type of tiller is connected to a two-way HDPE pipe faucet with a half-inch diameter. As the main pipe, a 2-inch-diameter PVC pipe is utilized so that the water volume and pressure can increase while distributing water to each row of HDPE pipes equipped with emitters. The land preparation and pipe installation process are described in (Delvion et al., 2021). Figure 2 depicts the visual appearance of the installed drip irrigation network.

Figure 2. Display of Drip Irrigation Network
The land area used in this study is ± 1,594.6 m2 or 0.1594 ha. The pipeline network consists of 26 loops, 186 outlets with 84 outlets with 1 emitter and 102 outlets with 2 emitters. The agricultural land is planted with coffee and tamarind plants. The distance between coffee plants and coffee plants is 2 meters and between coffee plants and tamarind plants is 4 meters while the distance between each tamarind plant is adjusted. The appearance of the drip irrigation network in Figure 2 is illustrated in Figure 3.  Figure 3 demonstrates that the constructed drip irrigation network is divided into two sections: the left section contains outlets with a single emitter for each outlet, while the right section contains outlets with two emitters for each outlet. To make it easier to determine the water flow equation for each pipe node, Figure 3 drip irrigation network is depicted in Figure  4 format.  Figure 4, the scheme of the water distribution network formed is a mixed scheme between a branched network and a network with loops. Using Equation (6) and Equation (7), 275 equations are obtained, divided into 249 node equations and 26 pressure reduction equations in the loop. For pipeline networks without loops (branched networks), the node equation is as follows: Next, by substituting the initial debit of the pipe water flow (Q1), which is 0.41 ml/sec into each node equation, the initial discharge for each node is obtained, as shown in Table 1. The next step is to form a Newton-Raphson multivariable iteration matrix as follows:

Emitter Water Debit Calculation
with ∆ as the column vector, {∆ }| =1,2,…, is the discharge correction vector for each water flow in the pipe, vector = { }| =1,2,…, is the column vector for flow discharge and the matrix J is the Jakobian matrix which is defined as follows: = [ ] with = = elements of the ith row and jth column, , = 1,2,3, … ,275.
In the previous step, it is known that there are 275 f equations and 275 Q variables, so the matrix ∆ has the dimension of 1 × 275, the matrix − has the dimension of 275 × 275 and the matrix has the dimension of 1 × 275, as follows: Replace the initial discharge value in each node equation, loop equation, and their derivatives with the initial discharge value. Due to the large size of the Jacobian matrix, its elements are not presented here, while those of the F matrix are shown in Table 2. Based on Table 5, it is known that ≈ 0, ∀ = 1,2, … ,275 . The iterative process for determining optimal ∆Q and Q is carried out using Equation (4) and Equation (5) and by using the help of Matlab R2021 software. From the calculation process, the discharge correction value (∆ ) is obtained as presented in Table 3. Since ≈ 0, ∀ = 1,2, … ,275, ∆Q ≈ 0, ∀ = 1,2, … ,275 is also obtained as shown in Table  3 so that the iteration is stopped. Then, equation (5) By using equation (10), the final discharge for each node is obtained. Based on the specification description of the drip irrigation network system, there are 84 outlets on the left side with 1 emitter and 102 outlets on the right side with 2 emitters. Table 4 presents the debit data for each emitter outlet on the left side and Table 5 presents the debit data for each emitter outlet on the right side. Since there are 2 emitters on the right side of the outlet, the amount of water discharge resulting from the calculation process needs to be divided by two, as shown in Table 4 and Table 5.