Evaluating the Pressure and Loss Behavior in Water Pipes Using Smart Mathematical Modelling

Abstract

Due to the constant need to enhance water supply sources, water operators are searching for solutions to maintain water quality through leakage protection. The capability to monitor the day-to-day water supply management is one of the most significant operational challenges for water companies. These companies are looking for ways to predict how to improve their supply operations in order to remain competitive, given the rising demand. This work focuses on the mathematical modeling of water flow and losses through leak openings in the smart pipe system. The research introduces smart mathematical models that water companies may use to predict water flow, losses, and performance, thereby allowing issues and challenges to be effectively managed. So far, most of the modeling work in water operations has been based on empirical data rather than mathematically described process relationships, which is addressed in this study. Moreover, partial submersion had a power relationship, but a total immersion was more likely to have a linear power relationship. It was discovered in the experiment that the laminar flows had Reynolds numbers smaller than 2000. However, when testing with transitional flows, Reynolds numbers were in the range of 2000 to 4000. Furthermore, tests with turbulent flow revealed that the Reynolds number was more than 4000. Consequently, the main loss in a 30 mm diameter pipe was 0.25 m, whereas it was 0.01 m in a 20 mm diameter pipe. However, the fitting pipe had a minor loss of 0.005 m, whereas the bending pipe had a loss of 0.015 m. Consequently, mathematical models are required to describe, forecast, and regulate the complex relationships between water flow and losses, which is a concept that water supply companies are familiar with. Therefore, these models can assist in designing and operating water processes, allowing for improved day-to-day performance management.

1. Introduction

In order to provide drinking water and sanitation services to the public and promote balanced ecological, economic, and social health in communities, both today and in the future, sustainable urban water systems are crucial. Many countries face the challenges of addressing population increase, water shortages, climate change, and the environment [1]. However, in industrialized countries, most urban water systems have been able to meet urban water and sanitation demand through the use of centralized structures [2]. Moreover, the urban system is mainly concerned with water treatment. The fundamental aim of a Water Distribution Network (WDN) is to supply enough clean water for users [3]. Maintaining a steady and secure water supply has become difficult for many cities due to increasing development and water shortages. A substantial quantity of water is wasted through distribution system pipes. Water leaking not only wastes water resources but also has high socioeconomic costs. It has been determined that water loss may be further decreased by increasing leak-detecting capabilities. These measures may support the environment by saving water and reducing both energy and greenhouse gas emissions [4].

Due to the exponential growth of both science and technology and the complexities therein, it is necessary to interact with mathematicians, scientists, and engineers in order to solve complex problems and help the industry achieve sophistication and economization. As we know that water is vital for humans, animals, plants, etc., it is necessary to correctly manipulate water during the day in order to prevent even one drop of water from being wasted. This is not only to conserve water but also to ensure that the water that is needed for people does not include any hazardous bacteria, chemical compounds, or undesired contaminants. Polluted water, such as with chemicals or microorganisms in drinking water, is dangerous to human health and can result in water-borne diseases [5,6,7]. The water distribution systems are mainly divided into three parts: (i) storing water, (ii) purification and water disinfection, and (iii) supply. Consequently, the detection and location of water leakages and their timely repair are highly crucial.

Most water distribution networks have losses, but the quantity of the water lost varies between regions and even within the same distribution system [8]. Furthermore, faulty pipes waste 7% of the water flowing into well-developed and controlled water distribution systems [9]. In some cases, more than half of the entire input volume into the network is lost due to leaking pipes in poorly monitored systems [10]. As a result, water that is lost from leaky pipes is a major difficulty for water utilities’ operating services, and it is recognized as an expensive problem that is associated with service interruptions, energy waste, and natural resource waste [11,12]. This also has implications for the quality of the drinking water, because leaks can bring microorganisms into the water distribution system under low-pressure conditions [13]. In addition, the financial impact of leaky pipes is significant and cannot be ignored. However, calculating the amount of intrusion using the traditional orifice equation, which overlooks the influence of the main-pipe flow velocity, may underestimate the volume of incursion and the public health risk. The study of correctly evaluating leakage and incursion rates through leaking water pipelines is gaining popularity [14]. The fact that leakage and incursion through pipe wall fractures do not always follow the conventional orifice equation (discharge flow varies with the square root of pressure differential across the orifice) is well known [15]. When the pressure difference across a pipe wall is positive (internal minus external pressure), leakage flow exits the pipe through any existing leak holes. However, the pressure differential might be negative in other cases, allowing incursion flow to enter the pipe. Because both leaks and incursions are undesirable in pipe systems, engineers and researchers in pipeline systems need a realistic model of their behavior [16].

This study expresses the smart mathematical models for water flow by taking both hydrostatic pressure and the pipe structure into account. The Osborne Reynolds experiment was also used to measure the flow characteristics of the liquid in the pipe and to calculate the Reynolds number for laminar, transitional, and turbulent flow. Furthermore, minor and major water losses were assessed in various pipes, and an orifice meter was used to monitor discharges in pipes. Again, the experimental data were compared to theoretical data that was generated from the smart mathematical model in this work.

2. Methods

2.1. Smart Mathematical Modelling of Water Flow

In most major cities of the developed country, public water services, as shown in Figure 1, are heading towards systems that can monitor the flows, pressure levels, reservoir levels, and so on, and communicate them to central stations in short intervals via text messages [17]. These data offer a variety of advantages for the management of water systems and researchers [18], including improving the overall water use and leak detection, identifying the location and scale of water stress, and analyzing water trends, all of which can aid in making decisions about water issues [19]. Moreover, models that simulate water usage can enhance the performance of water distribution networks [20]. The majority of existing water consumption models are built and used in a deterministic setting. The literature just recently published more realistic methods for assessing parameter and model uncertainty [21,22]. Understanding how fluid forces operate is necessary for designing the pipes in a real water network system. In order to construct a water network in urban areas, the amount of the resultant force and its center of pressure must be evaluated. This model is designed to show the hydrostatic pressure of water in the pipes by using the following equations.

The equations that were applied for the calculations are as follows:

(A)

Partially Submerged Body:

(1)

Hydrostatic Thrust (N):

$$F=\rho g\frac{B{d}^2}{2}$$

(1)

(2)

Experimental Center of Pressure (exp):

$$\mathrm{h}″=\frac{mgL}{F}$$

(2)

(3)

Theoretical Center of Pressure (M):

$$\mathrm{h}″=\mathrm{H}-\frac{d}{3}$$

(3)

(4)

Depth of Center of Pressure from free surface of water:

$$\mathrm{h}\prime = \mathrm{h}″+d-H$$

(4)

(5)

Depth of centroid of quadrant from free surface of water:

$$h=\frac{d}{2}$$

(5)

(B)

Fully Submerged Body

(1)

Hydrostatic Thrust (N):

$$F=\rho gBD\left(d-\frac{D}{2}\right)$$

(6)

(2)

Experimental Center of Pressure (exp):

$$\mathrm{h}″=\frac{mL}{\rho BD}\left(d-\frac{D}{2}\right)$$

(7)

(3)

Theoretical Center of Pressure (M):

$$\mathrm{h}″=\frac{\frac{D^2}{12}+{(d-\frac{D}{2})}^2}{\left(d-\frac{D}{2}\right)}+\left(H-d\right)$$

(8)

(4)

Depth of Center of Pressure from free surface of water:

$$\mathrm{h}\prime = \mathrm{h} ″+d-H$$

(9)

(5)

Depth of centroid of quadrant from free surface of water:

$$h=\frac{d}{2}$$

(10)

where: Density of water (ρ) = 1000 kg/m³, gravity number (g) = 9.81 m/s².

2.2. Smart Mathematical Modelling of Water Consumption with Periodic Time

A solution that mimics water usage as a function of real time is needed. Periodic functions are those that repeat their values at regular intervals, or periods [23]. The sine curve is more often used to imitate regular cycles, e.g., the number of hours of daylight each year, tones of music, human voice, breathing, or monthly electricity expenses.

The simplest basic form of a sine equation, in the function of time, is to describe periodic events such as water consumption as follows:

$$\mathrm{W}(t_i)=\sin (t_i)$$

(11)

where i refers to the data provision interval (time units) and W($t_i$) refers to the water consumption (volume) at any given moment.

This simple approach can be modified for further description by several additional factors, such as the range of the oscillation (A), which is the maximum difference from the center position of the function (volume units), and the angular frequency (ω), which stipulates how many oscillations take place in the unit time ($t_i$). The phase (φ) indicates where the oscillations occur in their cycle beginning at t = 0 (volume units), and a non-zero center amplifier, also called the offset value (ŵ), should be included in the function, which is, in fact, equivalent to the average water consumption of each cycle (v/t). The equation is rewritable as:

$$\mathrm{W}(t_i)=ŵ+\mathrm{A} \sin (\mathsf{\omega } t_i+\mathsf{\phi })$$

(12)

Due to the fact that the angular frequency (ω) is:

$$\mathsf{\omega }=\frac{2\pi }{T}$$

(13)

where T is wavelength or time (measured in units of time) and the Equation (3) in (2) is inserted provides:

$$\mathrm{W}\left(t_i\right)=ŵ+\mathrm{A} \sin [(\frac{2\pi }{T}) t_i+\mathsf{\phi }]$$

(14)

Equation (14) is now able to explain water use in urban areas with one oscillation per period in the function of time. If additional oscillations are shown in each period, the total of n waves propagating can be generalized by simply adding the second half of the equation (n-time as needed) proportionately to the daily flow cycles, as follows:

$$\mathrm{W}(t)=ŵ+{{\displaystyle \sum }}_{i=1 }^n[ A_i\sin (\left(\frac{2\pi }{T}\right)t+{\mathsf{\phi }}_i$$

(15)

Almost every day, flow opportunity may be represented using Equation (15). However, when more cycles are added to the equation, the value of the parameters diminishes. It does not straightforwardly calculate the amplitude of the sum of several sine curves, and it relies on whether or not the n periods are equal to each other. Therefore, three distinct options are available to be calculated:

Option 1: If times and phases are the same:

$$A_f={\displaystyle \sum }_{i=1}^n{A}_i$$

(16)

Option 2: if the periods are identical and the phases are different:

$$A_F=\sqrt{{\displaystyle \sum }_{i=1}^n{A}_i^2+2 {\displaystyle \sum }_{j=1}^n{A}_i{A}_j\cos \left({\mathsf{\phi }}_i-{\mathsf{\phi }}_j\right)}$$

(17)

Option 3: if the periods and phases are not the same:

$$A_F=\max \left[\left|\mathrm{W}\left(t_i\right)-ŵ\right|\right]$$

(18)

Option 3, which is the most likely situation, does not have a simple solution and must be mathematically calculated. In the worst-case situation of 23.18%, an error is created which would not be present if the AF is calculated using either Equations (6) or (7) [24]. This study is designed to demonstrate that the water consumption in our experiments can be compared and predicted. In order to depict water usage as a function of time, a model was built using regular functions. The model estimates how much water will be used based on the time of day. The following are the calculating formulas:

(1)

Flow

$$Q=\frac{Volume \left(V\right)}{Time \left(t\right)}$$

(19)

(2)

Area:

$$A=\pi {(\frac{Diameter \left(D\right)}{2})}^2$$

(20)

(3)

Velocity =

$$v=\frac{Flow \left(Q\right)}{Area \left(A\right)}$$

(21)

(4)

Reynolds number:

$$Re=\frac{Velocity \left(v\right) \times Diameter \left(D\right)}{Kinematic Viscosity \left(KV\right)}$$

(22)

2.3. Mathematical Modelling of Water Losses in Pipes

Even in stable conditions, pipes and pipe networks may suffer from capacity loss. For example, leakages might arise due to poor craftsmanship, damage, abrupt pressure fluctuations, corrosive effects, or maintenance deficiencies. In most situations, significant difficulties with harmful repercussions for the emergence of leaks can arise. Leaks in drinking water or wastewater networks, for example, generate a lot of problems: money lost, transportation disruption, and damage to adjacent infrastructure.

In order to reduce these negative impacts, techniques for detecting leaks in pipelines are desirable. Flow direction indicators, tracer gases, subsurface radar, earth resistivity changes, infrared spectroscopy, microphonics, and odorant and radioactive tracers are among the most common experimental procedures utilizing field testing that are documented in the literature at this time. These approaches are costly and time-consuming, and we need to concentrate on mathematical modeling to solve these problems. This will surely save time and difficulty when performing outdoor tests [25]. A mathematical model for forecasting the losses in pipes was created based on transient calculations [26,27,28]. As the flow is turbulent in most industrial applications, the current study seeks to provide a mathematical model that allows us to find the site of the leak in pipes that are transporting incompressible liquids, or moving in laminar systems or turbulent regimes. The I model aims to offer a theoretical approach to the situation of pipeline holes (Figure 2).

Figure 2 shows a water pipe with a hole. The variables that impact the problem of the steady state are:

At the entrance area:

The pipeline inlet discharge (Q1)

Static head in the inlet area (h1)

At the exit area

The release from the pipe (Q2)

The outlet static head (h2)

At the position of the hole

The flood flow from the hole (Qx)

Static head from the hole initial position upstream (h_x1)

Static head from the hole downstream (h_x2)

C_d A_x: The hole’s real area (A_x) and the coefficient of discharge (C_d).

The location of the leak as measured from the intake section (X)

The total energy per unit weight, shown by the variable E in Figure 3, is the sum of the static and dynamic heads:

$$E_i=h_i+V_1^2 /2g$$

(23)

where i = 1 at inlet 2, X1 and X2 just before the hole and after the hole, respectively.

The above-mentioned set of variables contains nine unknowns. If the inlet and exit discharge and pressure are both measured, 5 equations are needed to solve the problem and produce a well-posed mathematical model.

The following are the five equations that explain the mathematical model that provides the solution for this issue:

(i) Equation of continuity:

$$Q_1-Q_2=Q_X$$

(24)

(ii) The friction losses in the length X of the pipe upstream from the hole can be represented as follows:

$$h_1-h_{x1}=f_1 \frac{X}{D} \frac{Q_1^2}{2g A^2}=k_1 X Q_1^2$$

(25)

where A is the cross-sectional area of a circular pipe of diameter D and length L.

K₁ is the hydraulic resistance per length unit ($\frac{{\mathrm{f}}_1}{\mathrm{D}2{\mathrm{g} \mathrm{A}}^2}$).

(iii) The friction losses in the section of pipe L-X downstream from the hole can be represented as follows:

$$h_{x2}-h_2=f_2 \frac{L-X}{D} \frac{Q_2^2}{A^{2}2g}=k_2 \left(L-X\right)Q_2^2$$

(26)

where:

The friction factor $f_2$ corresponds to the flow velocity V₂.

K₂ represents the hydraulic resistance per unit length.

(v) The following equation would be used to determine the discharge flow through the hole:

$$Q_x=C_d{A}_x\sqrt{2g{E}_{x1}}$$

(27)

(vi) Finally, using the entire energy balance on the system, the total energy lost in the hole H is calculated using the following equation:

$$\mathrm{H}=Q_1{E}_1-Q_2{E}_2-Q_1 \left(E_1-E_{x1}\right)-Q_2 \left(E_{x2}-E_2\right)$$

(28)

For an incompressible flow, the following equation shows that the overall energy lost in the hole is equal to the overall energy input minus the sum of the overall energy output plus the overall energy lost in the pipe before and after the hole, respectively.

The equation, after algebraic manipulation:

$$\mathrm{H}=Q_1{E}_{x1}-Q_2{E}_{x2}$$

(29)

When the stream flows into the environment

$$\mathrm{H}=(Q_1-Q_2) V_x^2/2g$$

(30)

where

$$V_x=\frac{Q_x}{C_d{A}_x}$$

(31)

is the fluid velocity that comes out of the hole. Replacing V_x as a per-unit discharge efficient flow area via the hole, we can simply demonstrate that:

$$H=C(Q_1-Q_2{)}^3$$

(32)

where

$$C=1/2(C_d{A}_x{)}^2$$

(33)

Noting the constant 1/2g A² by equating the two formulas for H, and expressing $E_{x1}$ and $E_{x2}$ in terms of the corresponding sums of static and dynamic heads

By B, one gets

$$Q_1{h}_{x1}-Q_2{h}_{x2}=\left(Q_1-Q_2\right)\left[\left(C-B\right)Q_1^2-\left(2C+B\right)Q_1{Q}_1+\left(C-B\right)Q_2^2\right]=(Q_1-Q_2)\mathsf{\lambda }$$

(34)

where

$$\mathsf{\lambda }=\left(C-B\right)Q_1^2-\left(2c+B\right)Q_1{Q}_2+\left(C-B\right)Q_2^2$$

(35)

In the preceding part, a number of equations that describe the physical phenomena were developed. Now, we want to discover the formula to forecast the losses in terms of the observed quantities at the pipe intake and outlet, most of which are $h_1$, $h_2$, $Q_1$, and $Q_2$. For this purpose, the equations hx1 and hx2 in Equations (24) and (25), respectively, are replaced with the values in Equation (20) and X resolved as follows:

$$X_c=\frac{\left(h_1-h_2\right)-k_{2}L{Q}_2^2+\left(Q_1/Q_1-1\right)\left(h_1-\lambda \right)}{k_1{Q}_1^2{Q}_1/Q_2 +k_2{Q}_2^2}$$

(36)

where $X_c$ is the calculated value of X.

3. Results

3.1. Application of Smart Mathematical Modelling of Water Flow

Hydrostatic Pressure

The force generated by a liquid’s pressure loading acting on submerged surfaces is known as hydrostatic pressure [29]. An understanding of how fluid forces work is required for the design of the model’s components [30]. The size of the resultant force and its center of pressure must be determined for such a design. Fluid mechanics begins with the calculation of the hydrostatic force and the position of the center of pressure. The pressure center is the location on the submerged surface where the resulting hydrostatic pressure force is applied. Furthermore, Figure 4A illustrates the experiment’s final equipment configuration, whereas Figure 4B depicts the apparatus dimensions. An empty flotation tank was put on a hydraulic bench, and the screw on the adjustable feet was adjusted until the built-in circular spirit level showed that the tank was level in both planes. The balancing arm was then placed on the knife-edge, and the arm was examined and assured to be free to swing. An empty weight hanger was also put in the groove at the end of the balancing arm. Finally, the counterbalance weight was adjusted until the balancing arm was horizontal and the center index mark on the beam level indicator was checked.

First, the width of the quadrant face (B) and the height of the quadrant face (D) were measured. Then, as indicated in

Table 1. Technical data.

L, Length of Balance	0.29 m	Distance from weight hanger to the pivot point
H, Quadrant to Pivot	0.21 m	Base of quadrant face to pivot
D, Height of Quadrant	0.10 m	Height of vertical quadrant face
B, Width of Quadrant	0.08 m	Width of vertical quadrant face

, the horizontal distance between the pivot point at the right hanger (L) and the vertical distance between the pivot and the base of the quadrant (H) was measured. Following that, the arm was positioned and balanced, and the water was added until the flotation tank was full. However, the plane surface was assumed to be partially submerged, and weights were progressively added to the weight hanger in order to bring the arm back into its equilibrium position.

Table 2. Experimental vs theoretical data for partially and fully submerged.

	Partially Submerged			Exp.	Theoretical	Exp.	Theoretical
No.	Mass, M (kg)	Depth of Immersion	Hydrostatic Force, F (N)	Depth of Center of Pressure from Pivot Point = h″ (m)	Depth of Center of Pressure from Pivot Point = h″	Depth of Center of Pressure from Free Surface of Water = h′ (exp)	Depth of Center of Pressure from Free Surface of Water = h′ (m)
1	0.02	0.02	0.17	0.25	0.20	0.07	0.01
2	0.05	0.04	0.66	0.21	0.20	0.04	0.03
3	0.07	0.05	1.03	0.19	0.19	0.04	0.04
Fully Submerged				Exp.	Theoretical	Exp.	Theoretical
1	0.31	4.59	0.18	0.17	0.08	0.08	0.08
2	0.36	5.53	0.18	0.17	0.09	0.09	0.09
3	0.45	7.36	0.17	0.17	0.10	0.11	0.11

shows the weight of their mass and the depth of the water surface at which equilibrium was attained. Finally, the flotation tank’s water level was raised until the plane surface was completely immersed.

In order to prevent water from draining during the experiment, the drain cock was blocked. Water was progressively poured into the tank, and a weight was placed on the balance pan until the beam reached equilibrium. The above procedures were repeated for each weight increase until the water level reached the top of the quadrant end face. Table 2 depicts the water level on the quadrant, as well as the measurement and calculations for experimental and theoretical modeling

The hydrostatic force on the quadrant grew as the depth of immersion increased, as seen in Figure 5A–D. These figures confirm the idea that as the depth of immersion in a system grows, so does the hydrostatic force. Interestingly, the difference in data between the partially submerged Figure 5A,B and the totally submerged Figure 5C,D resulted in a power relationship, whereas the fully submerged was likely linear. This means that the hydrostatic force in the partially immersed quadrant would grow at a faster pace due to its power relation, but the hydrostatic force in the completely submerged quadrant would increase in a linear connection with the depth of immersion. Furthermore, Figure 5 supports the conclusion that the depth of the center of pressure dropped as the depth of immersion in the system grew. When comparing the subfigures of Figure 5, it was discovered that the depth of the center of pressure in the completely submerged quadrant dropped with a power relation, whereas the partially submerged quadrant decreased linearly. This explains why the depth of the totally immersed quadrant would decrease more quickly. As a result, for both the fully immersed and partially submerged quadrants, there is a rising linear correlation between hydrostatic force and immersion depth. This is due to the fact that as depth increases, greater fluid weight exerts a downwards pull from above, raising the hydrostatic pressure at a given depth. The depth of the center of pressure, on the other hand, appears to decrease as the depth of immersion in the system increases. In practice, however, the concept of hydrostatic pressure may be employed and used in water-control structures such as leaks and gates in order to identify the position and quantity of the water pressure acting on the structures [31,32].

3.2. Application of Smart Mathematical Modelling of Water Consumption with Periodic Time

Osborne Reynolds Pipe Flow

The Osborne Reynolds experiments determined the flow characteristics of the liquid in the pipe and computed the Reynolds number for each flow condition [33]. Fluid flows are classified into three types: laminar, turbulent, and transitional, as shown in Figure 6. The Reynolds number has no dimensions and may be computed using the proper formula [34]. The dye was injected into the dye reservoir at the flow during this experiment in order to visualize the flow over time. Flow rates may be calculated by collecting the liquid that flows out of the Osborne Reynolds device into a beaker and recording the time that was required to fill up to a particular volume. Using the flow control valve, one can create laminar, turbulent, and transitional flows by changing and modifying the water flow rate. We calculated the range for laminar, transitional, and turbulent flow based on the data we obtained as shown in

Table 3. Calculations for Reynolds Number (Re).

	Volume V M3	Time t (s)	Diameter of the Pipe D (m)	Discharge Q m3/s	Area A m2	Velocity ν m/s	Kinematic Viscosity v m2/s	Re
Laminar Flow	0.0001	13.20	0.01	75 × 10−7	785 × 10−7	0.096	8917 × 10−10	1081
	0.0002	17.81	0.01	11 × 10−6	785 × 10−7	0.143	8917 × 10−10	1604
Transitional Flow	0.0003	16.69	0.01	18 × 10−6	785 × 10−7	0.229	8917 × 10−10	2567
	0.0005	22.86	0.01	22 × 10−6	785 × 10−7	0.279	8917 × 10−10	3125
Turbulent Flow	0.0005	15.00	0.01	33 × 10−6	785 × 10−7	0.425	8917 × 10−10	4762
	0.0008	24.10	0.01	33 × 10−6	785 × 10−7	0.423	8917 × 10−10	4742

.

The apparatus was set up as indicated in Figure 7, and the diameter of the pipe and the temperature of the room were measured. The bench control valve was opened, the apparatus flow control valve was gradually opened, and the system was allowed to fill with water. When the water level in the head tank reached the overflow tube, the bench control valve was adjusted in order to give a slow overflow rate. The bench control valve was adjusted, and the apparatus flow control valve was set to a steady trickle in order to reduce the overflow rate. It took a few minutes for the flow to settle. The dye control valve was set up to provide slow flow with obvious dye indication. A precise and well-defined line was identified. The flow rate was estimated after the dye control valve was closed.

Furthermore, fluid flow is studied using the Reynolds number. The Reynolds number assesses whether a fluid flow is laminar or turbulent. A transition flow is a fluid flow that exists between laminar and turbulent. In general, a fluid with laminar flow has a Reynolds number less than 2000, whereas fluids with turbulent flow have Reynolds numbers greater than 4000. The transition flow ranges between 2000 and 4000 [35]. In the experiment, it can be shown that both trials, while testing for laminar flow, had Reynolds numbers of less than 2000. On the other hand, when testing for transitional flows, both Reynolds numbers were determined to be in the range of 2000 to 4000. When testing for turbulent flow, the Reynolds number for both trials exceeded 4000. This demonstrates that the theory of fluid flow prior to the experiment, which was based on the Reynolds number, is the same as the calculated values based on the Reynolds formula. As a result, the characteristics and kinds of flow profiles in the fluid of water pipes are essential variables in the fluid flow process. The Reynolds number indicates the fluid flow regime in pipes. The influence of any deviation in the dye density will be noticed against the fluid flow mechanism if the vertical flow is used [36]. Fluid is delivered flexibly from the pipe through each available space and the speed of fluid distribution is affected by the flow profile. The flow rate is low if it is laminar; on the other hand, a turbulent flow indicates a high fluid velocity in the pipe [37]. The friction coefficient value (λ) will rise as the diameter (D) of the test pipe increases. In addition, the coefficient of friction (λ) decreases as the diameter (D) of the test pipe decreases. As a result, the diameter of the pipe to be examined is one of the elements that affects the flow profile in a real network system.

3.3. Application of Smart Mathematical Modelling of Water Losses in Pipes

3.3.1. Orifice Meter

An orifice is described as an aperture or a mouth of any shape or size through which fluids are released in the form of a jet, and is often found in a pipe or on the bottom sidewall of a water tank or container, as shown in Figure 8. An orifice meter measures the discharges in pipes, and is able to determine the flow measurement of a pipe by using Bernoulli’s equation [38]. Bernoulli’s equation connects a fluid’s pressure to its velocity, where the velocity and pressure of a liquid are inversely related. According to the rule of conservation of energy, as fluid velocity increases, fluid pressure decreases in order to conserve mechanical energy [39].

In this experiment, an orifice will be utilized to quantify the discharge in a pipe flow setting. The discharge through an orifice meter may be calculated as follows:

$$Q=C_{d}A\sqrt{2gH}$$

(37)

where

Q = discharge

Cd = coefficient of discharge of the orifice meter (for this experiment, the value is Cd 0.67)

H = the difference in the piezometer head between H1 and H2

From the energy equation,

$$\frac{P_1}{{\rm Y}}+\frac{v_1^2}{2g}+Z_1=\frac{P_2}{{\rm Y}}+\frac{v_2^2}{2g}+Z_2+h_L$$

(38)

Therefore,

$$H=\frac{P_1}{{\rm Y}}-\frac{P_2}{{\rm Y}}$$

(39)

For this experiment, the diameter of the pipe (upstream of the orifice) was 28.5 mm, whereas the diameter of the aperture was 18.5 mm. The flow was measured using an orifice meter in this experiment. The obtained results were compared to known values for the discharge coefficient. Cd, or the usual discharge coefficient, was 0.67, and is indicative of the amount of time required to collect 20 L of water. As a result, the flow rates for a half-opened and fully opened valve were calculated and are summarized in

Table 4. Calculation of the flow rate in orifice meter.

	Halve Open Valve	Fully Open Valve
Experimental (Q)	2.53 × 10−4 m3/s	3.99 × 10−4 m3/s
Theoretical (Q)	2.52 × 10−4 m3/s	4.00 × 10−4 m3/s
Percentage error (%)	2.35	0.26

.

In a half-opened valve, the discharges via orifice that were derived from the equation are lower than the discharge that was obtained directly from measurement, with a percentage error of 2.37 percent. Furthermore, when the valve is completely opened, the discharges through the orifice that were derived from the equation are greater than the discharge that was obtained directly from measurement, with a percentage error of 0.25 percent. As a result of these errors, the findings deviate slightly from what was expected. Observing the percentage error, the error in this experiment may be attributed to the instrument’s lack of precision, leaks, and occasional wear and tear, which can impact the results. The differences in the piezometer head between sections 1 and 2 for the flow rate of a fully opened valve and half-opened valve are summarized in Table 5.

Table 5. The difference in the piezometer head between sections 1 and 2 in half and fully opened valve.

Half Opened Valve
Piezometer head at section 1, m	H1	0.529
Piezometer head at section 2, m	H2	0.437
The volume of water, L	V	20
Time, s	t	87
The difference in the piezometer head between sections 1 and 2, m	H	0.092
Fully Opened Valve
Piezometer head at section 1, m	H1	0.763
Piezometer head at section 2, m	H2	0.505
The volume of water, L	V	20
Time, s	t	49
The difference in the piezometer head between sections 1 and 2, m	H	0.258

Table 5. The difference in the piezometer head between sections 1 and 2 in half and fully opened valve.

Half Opened Valve
Piezometer head at section 1, m	H1	0.529
Piezometer head at section 2, m	H2	0.437
The volume of water, L	V	20
Time, s	t	87
The difference in the piezometer head between sections 1 and 2, m	H	0.092
Fully Opened Valve
Piezometer head at section 1, m	H1	0.763
Piezometer head at section 2, m	H2	0.505
The volume of water, L	V	20
Time, s	t	49
The difference in the piezometer head between sections 1 and 2, m	H	0.258

where:

• Cd (Coefficient of discharge) = 0.67
• Diameter of the pipe (upstream) = 28.5 mm
• Diameter of the orifice = 18.5 mm
• Area cross-section of the orifice = $\frac{\pi d^2}{4}=$ 2.69 × 10⁻⁴ m²

An orifice meter was used to determine the pressure differential between the upstream and downstream pipes, and the water discharge rate, or flow rate. Because of the increased pressure, the flow rate increases as the height difference grows. Several variables influence head loss, the majority of which are connected to energy loss due to friction. Even though losses in these areas are referred to as minor losses, they can occasionally be larger and more consequential than major losses. However, we learned from this experiment that even if the pipes or connections are the same, we cannot generalize the coefficient of loss since many other factors might impact the flow of fluid through the pipes. Orifice meters are commonly used in engineering practices to monitor fluid flow in pipelines and reservoirs. An orifice meter, for example, is positioned on the portion of the flow that enters the intake or storm drain. The velocity and discharge will then be measured using the orifice meter. Another example of utilizing an orifice meter is the dam bottom line [40]. The velocity and discharge coefficients are required to forecast flow rates from orifices [41,42]. Moreover, according to the literature review findings, a significant amount of research has been performed in the design and assessment of orifice meters. Sondh et al. [43], for example, were interested in developing and designing variable area orifice meters. They utilized a variable area orifice meter to show the flow rate as a linear displacement of an asymmetrical body positioned concentrically downstream of an orifice within a constant area duct. They used three symmetrical body forms to construct a variable area orifice meter: a frustum of a cone, a frustum of a cone with a hemispherical base, and a frustum of a cone with a hemispherical base and a parabolic apex downstream of an orifice. They conducted comparable trials at various areas of the symmetrical bodies in order to assess the performance of the variable area orifice meters. They showed that, given a constant pressure differential, the frustum of a cone with a hemispherical base and a parabolic apex generates a nearly linear variation in flow rate with the position. Takahashi et al. [44] also investigated the cavitation properties of a restriction orifice in two separate experiments.

The spatial distribution of cavitation shock pressure in a pipe downstream of limitation orifices was the subject of their first experiment. The examination of butterfly valve throttling to cavitation in a multi-perforated orifice installed pipe was their second experiment. Based on the findings of their investigation into cavitation shock pressure, it can be inferred that the maximum shock pressure increases noticeably with decreasing cavitation number independent of orifice type. For a cone-type orifice, the maximum shock pressure is at its lowest, and at its highest for a single hole orifice. Between these two is the multi-perforated orifice. When the multi-perforated orifice is positioned at 1D downstream of the butterfly valve, the results of their experiment regarding the occurrence of cavitation due to the interference of the butterfly valve indicated that cavitation occurs at a reasonably high cavitation number. Because of the closed orifice installation, the butterfly valve throttling promotes cavitation at the multi-perforated orifice. As a result, employing an orifice meter makes it much easier for engineers to perform their tasks.

3.3.2. Losses in Pipe

In order to transport a volume of liquid through a pipe, a certain quantity of energy is necessary [45]. To induce the liquid to move, there must be differences in energy or pressure [46]. A part of that energy is wasted due to flow resistance. This resistance to flow is known as significant head loss, and it is caused by fluid friction between the water and the pipe. The loss is produced by viscosity and is proportional to the wall shear stress. The equation for the head loss along the length of a straight pipe with a constant diameter is:

$$h_L=F \frac{L}{D}\frac{V_{avg}^2}{2g}$$

(40)

where $F$ is the Darcy–Weisbach friction factor, which is a function of Reynolds number of the flow and the roughness of the internal surface of the pipe.

The energy loss that happens when a fluid contacts fittings such as bends, inlets, outlets, expansions, or contractions, on the other hand, is less significant when compared to frictional effects and is hence referred to as a small loss. In order to address variations in flow direction, as well as changes in flow area due to the pipe system’s complexity and the number of fittings utilized, the head loss coefficient ($K_L$) is calculated experimentally as a rapid way to calculate small losses. Minor losses are stated as:

$$h_L=K_L\frac{V^2}{2g}$$

(41)

where $K_L$ = loss coefficient, depending on the type of fitting

The characteristics and consequences of major and small losses apply to a wide range of man-made and natural phenomena, from large-scale industrial systems to microscopic biological systems, and are important factors to consider when developing fluid transport systems. For example, the design of a nuclear reactor necessitates the careful construction of such systems and the losses involved, since the geometry of the pipe and coolant loss play a significant influence on the overall efficacy and safety of the system.

As shown in Figure 9, piezometer tubes were installed on the manometer and used in the experiment to measure friction in all of the pipes. When the supply was turned on, the pipe’s manometer ports were connected to the pipe. The bubbles in the pipe were eliminated by applying pressure to the flow. After the bubbles were eliminated entirely, the flow rate was determined by collecting 5 L after the drain valve was closed. The amount of time spent was recorded. The pressure heads before and after a fluid encountered abrupt changes in pipe diameter or flow direction, H1 and H2, were monitored using Piezometer tubes in the experiment. As a result, according to

Table 6. The losses in pipes (major and minor losses).

		D	L	H1	H2	H	V	t
Major losses		30 mm	50 cm	44 cm	41.7 cm	4.7 cm	5 L	38 s
		20 mm	80 cm	55.2 cm	54.2 cm	1 cm	5 L	75 s
Minor Losses	Fitting	20 mm and 30 mm		42.8 cm	42.2 cm	0.6 cm	5 L	37 s
	Bending			42.7 cm	41.7 cm	1 cm	5 L	63 s

, the major losses in the 30 mm diameter pipe were 0.25 m, whereas the major losses in the 20 mm diameter pipe were 0.01 m. However, the minor losses in fitting were 0.005 m, while the minor losses in bending were 0.015 m.

According to K_L (Figure 9), the difference between those values is due to the fact that there are no extra losses via the pipe. Because of the intricacy of the flow in numerous fittings, the experiment is larger than predicted. When water flows through the pipeline, the pressure is no longer simply determined by the height difference between the lowest and highest point. Friction between the water and the pipe causes a pressure decrease. The diameter of the pipe affects the head loss inside the system. The greater the friction with a continuous flow of water, the smaller the pipeline’s diameter [47]. The next aspect to consider is the pipe material, because the smoother the inner surface of the pipe, the smaller the head loss [48]. PVC pipe, for example, has a smaller head loss than steel pipe under comparable circumstances because it is smoother [49]. The length of the pipe is the third element influencing head loss [50]. The higher the head loss, the longer the pipeline. This loss is inversely proportional to length. The velocity of the water in the pipe is another element that influences head loss [51]. The head loss due to friction increases as the flow rate of water increases.

The number of fittings or bends in the pipeline is another element that affects head loss [52]. A straight-line pipeline will have less head loss than a pipeline with fittings or bends of the same length. As a result of this experiment, it is possible to conclude that head losses in pipes may be classified into two types: major head losses and minor head losses. Major head losses are head losses or pressure losses that are caused by friction in pipes and ducts, whereas small head losses are caused by components in the pipes such as fittings, bends, and valves. When a fluid travels over abrupt changes in pipe diameter or flow direction, the pressure heads can be detected with a piezometer. For a fixed flow rate, the Darcy–Weisbach equation states that the head loss decreases with the inverse fifth power of the pipe diameter. On a regular basis, it is critical to confirm that a pipe is acceptable for a certain piping installation.

4. Conclusions

A mathematical model was developed to describe the water flow system. Moreover, the model was demonstrated to be realistic in identifying losses in pipelines. The solution method was based on an iterative methodology that led to the measurement of the minor and major losses of water. Knowing the hydrostatic pressure of a certain water-related structure will also aid in structural design to guarantee that the water pressure force exerted does not cause the structure to fail. Moreover, all of the calculations fell within the parameters indicated by Reynolds number, allowing us to infer that the theory is important and relevant in real life. An engineer may be able to enhance a pipe’s discharge capacity while minimizing head losses per unit length of the pipe. As a result, when an engineer needs to enhance pipe production, these factors will be considered.