The study considers both uniform and varying depths. Uniform depth applies to the straight sections of the culvert, while varying depth is analyzed near the BI, where slope and orifice effects cause disruptions. The critical slope depends on the channel dimensions and flow conditions. Subcritical flow dominates the studied system, but an in-depth analysis of Froude number and the critical slope is recommended for future studies [5] .

The scenarios focus on mild slopes, as seen in the constructed channel. The steep rise near BI disrupts flow efficiency, creating challenges in Case 2. The study primarily examines uniform flow but acknowledges non-uniform flow effects near the BI orifice. These effects could be analyzed further using simulation tools in future studies.

All cases satisfy the Manning equation as the hydraulic radius, slope, and roughness coefficient are used to compute the velocity and flow rate. For instance, Case 2 incorporates the slope and adjustments from construction to calculate the flow rate accurately. The Manning equation is derived from the Chézy equation. The Chézy coefficient can be expressed by linking both equations. The Manning equation refines the Chézy equation by explicitly incorporating channel roughness and is used here for more practical applications [6] .

Manning’s Equation

Manning’s formula describes the flow velocity (v) of a liquid in open channels as a function of channel characteristics and slope [7] .

$v=\frac{1}{n}{R}^{2/3}{S}^{1/2}$ (1)

Mathematical Variables:

v: Average flow velocity (m/s)

n: Manning’s roughness coefficient (dimensionless)

R: Hydraulic radius (m), defined as:

$R=\frac{A}{P}$ (2)

where A is the cross-sectional flow area (m²), and P is the wetted perimeter (m).

S: Channel slope (m/m), representing the energy grade line or bottom slope.

Application:

The flow discharge (Q) in an open channel can be obtained by combining the velocity v with the flow area A:

$Q=v\cdot A=\frac{1}{n}{R}^{2/3}{S}^{1/2}A$ (3)

This formula is widely used in civil engineering, particularly in the design and analysis water systems, as well as in natural rivers and other watercourses. It helps engineers determine flow quantities necessary for various calculations, including flood forecasting, irrigation scheduling, and infrastructure design.

Another fluid mechanic formula to be introduced is orifice equation. The orifice equation is used to calculate the flow rate of fluid passing through an orifice, which is a small hole or opening in a container or barrier. This equation is a fundamental part of fluid dynamics and is especially useful in various engineering applications, including measuring the flow rates in tanks, pipes, and other fluid conveyance systems.

The basic form of the orifice equation can be derived from Bernoulli’s principle and the continuity equation, and it is typically expressed as follows:

Orifice Equation

The orifice equation models the discharge (Q) through a small opening, where the velocity of flow is derived from Bernoulli’s principle. It is given as:

$Q=C_{d}A\sqrt{2gh}$ (4)

Mathematical Variables:

Q: Volume flow rate (m³/s)

C_d: Discharge coefficient (dimensionless), accounts for energy losses due to friction and geometry

A: Cross-sectional area of the orifice (m²), where A = πr² for circular openings

g: Gravitational acceleration (9.81 m/s²)

h: Head or fluid height above the orifice centerline (m).

Application:

The velocity of the fluid (v) through the orifice can be expressed as:

$v=\sqrt{2gh}$ (5)

Thus, the flow rate becomes:

$Q=C_{d}Av=v=\sqrt{2gh}$ (6)

Figure 5 shows the original design where the Bandau Intake (BI) depth is greater than the Infiltration Gallery (IG) depth. Using Manning’s equation, the velocity and flow rate are calculated based on the assumed hydraulic radius, slope, and roughness coefficient.

Determine the velocity of water along 30 m long of culvert:

Hydraulic Radius, R—Assuming full flow in a square culvert, the hydraulic radius can be approximated as:

$R=\text{Area}/\text{Wetted Perimeter}$

$R=\left(1.2\times 1.2\right)/\left(1.2\times 4\right)=1.44/4.8$

$R=0.3$

Slope Calculation, S—Assuming the 0.3 m drop is across the entire length of the culvert

$S=\text{Gradient}/\text{Distance}$

$S=0.3\text{m}/30\text{m}$

$S=0.01$

Manning’s roughness coefficient, n for concrete—0.013

By using Manning formula:

$v=\frac{1}{n}{R}^{2/3}{S}^{1/2}$

Simplified the formula:

$V=1/n\times R^{0.67}\times S^{0.5}$

$V=1/0.013\times {0.3}^{0.67}\times {0.01}^{0.5}$

$V=3.43\text{m}/\text{s}$

Determine the water flow rate (Q) along 30 m of culvert:

The flow rate (Q) is the product of the velocity and the cross-sectional area.

Culvert cross-sectional area, A = 1.2 m × 1.2 m = 1.44 m²

$Q=A\times V$

$Q=1.44{\text{m}}^2\times 3.43\text{m}/\text{s}$

$Q=4.94{\text{m}}^3/\text{s}$

Therefore, the flow rate along the 30 m culvert is 4.94 m³/s.

Figure 6 shows the constructed channel incorporates a slope and an orifice to control flow rate. Adjustments in elevation and slope near the BI were considered, and calculations of flow velocity and rate were performed using the Manning and orifice equations [8] .

Note:

1) The culvert drops by 0.3 meters at the 25-meter mark due to an obstruction.

2) The culvert then rises to 1.8 meters over the last 5 meters to reach Bandau Intake.

3) An orifice or opening with an area A₀ of 0.16 m² is installed at Bandau Intake inlet.

4) The cross-sectional area of the 25 m length of culvert is, A = 1.44 m².

Calculate the Velocity at the 25-meter Mark (V_25m)

Hydraulic Radius, R—Assuming full flow in a square culvert, the hydraulic radius can be approximated as:

$R=\text{Area}/\text{Wetted Perimeter}$

$R=\left(1.2\times 1.2\right)/\left(1.2\times 4\right)=1.44/4.8$

$R=0.3$

Slope Calculation, S—Assuming the 0.3 m drop is across the entire length of the culvert.

$S=\text{Gradient}/\text{Distance}$

$S=0.3\text{m}/25\text{m}$

$S=0.012$

Manning’s roughness coefficient, n for concrete—0.013

Manning Formula:

$v=\frac{1}{n}{R}^{2/3}{S}^{1/2}$

Simplified the formula:

$V_{25\text{m}}=1/n\times R^{0.67}\times S^{0.5}$

$V_{25\text{m}}=1/0.013\times {0.3}^{0.67}\times {0.012}^{0.5}$

$V_{25\text{m}}=3.76\text{m}/\text{s}$

Calculate the Flow Rate at the 25-meter Mark (Q_25m)

With the velocity, we can determine the flow rate:

$Q_{25\text{m}}=A\times V_{25\text{m}}$

$Q_{25\text{m}}=1.44{\text{m}}^2\times 3.76\text{m}/\text{s}$

$Q_{25\text{m}}=5.4{\text{m}}^3/\text{s}$

Account for Elevation Rise Over the Last 5 Meters

The water rises by 1.8 meters over the last 5 meters to reach Bandau Intake, so the total effective head $h_{\text{eff Bandau Intake}}$ just before the orifice is:

$h_{\text{eff Bandau Intake}}=h_{\text{eff 25m}}-1.8\text{m}$

$h_{\text{eff Bandau Intake}}=0\text{m}-1.8\text{m}$

$h_{\text{eff Bandau Intake}}=-1.8\text{m}$

There is a negative value, which means the water has to rise, and we’d expect a reduction in kinetic energy converting to potential energy to overcome the elevation.

Calculate the Velocity and Flow Rate Through the Orifice

Velocity at the Orifice (V_orifice): The velocity of water through the opening or orifice can be calculated using Bernoulli’s equation, but for orifices, we also typically introduce a discharge coefficient C_d to account for the reduction in velocity due to the vena contracta effect where the flow streamlines converge, causing the actual jet area to be less than the orifice area.

$V_{\text{oriffice}}=C_d\sqrt{2\times g\times h_{\text{eff}}}$

The discharge coefficient C_d is for sharp-edged orifices, and it typically ranges from 0.6 to 0.7. Will use the maximum value of 0.7.

We would then calculate the velocity through the orifice, assuming there’s enough head to drive the flow:

$V_{\text{oriffice}}=C_d\sqrt{2\times g\times h_{\text{eff}}}$

$V_{\text{oriffice}}=0.7\sqrt{2\times 9.81\times 1.8}$

$V_{\text{oriffice}}=4.16\text{m}/\text{s}$

Due to the effective head is −1.8 m, therefore the V_{orifice(opening)} is −4.16 m/s.

$Q_{\text{orrifice}}=A_{\text{orrifice}}\times V_{\text{orrifice}}$

$Q_{\text{orrifice}}=0.16{\text{m}}^2\times \left(-4.16\text{m}/\text{s}\right)$

$Q_{\text{orrifice}}=-0.67{\text{m}}^3/\text{s}$

Hence,

Total Flowrate ( $Q_{\text{25m}}+Q_{\text{orrifice}\text{5m}}$)

$Q_{\text{total}}=5.4{\text{m}}^3/\text{s}+\left(-0.67{\text{m}}^3/\text{s}\right)$

$Q_{\text{total}}=4.73{\text{m}}^3/\text{s}$

Therefore, the flow rate along the 30 m (25 m + 5 m rise) culvert is 4.73 m³/s.

Figure 7 shows the scenario of case 3. This hypothetical scenario assumes the BI opening has unrestricted flow. Without an orifice, significant inefficiencies and backflow were observed. Flow rate and velocity were calculated using Bernoulli’s principle and Manning’s equation to illustrate the impact of unregulated flow [9] .

Note:

1) The culvert drops by 0.3 meters at the 25-meter mark due to an obstruction.

2) The culvert then rises to 1.8 meters over the last 5 meters to reach Bandau Intake.

3) The cross-sectional area of the culvert is 1.44 m².

Calculate the Velocity at the 25-meter Mark (V_25m)

Hydraulic Radius, R—Assuming full flow in a square culvert, the hydraulic radius can be approximated as:

$R=\text{Area}/\text{Wetted Perimeter}$

$R=\left(1.2\times 1.2\right)/\left(1.2\times 4\right)=1.44/4.8$

$R=0.3$

Slope Calculation, S—Assuming the 0.3m drop is across the entire length of the culvert

$S=\text{Gradient}/\text{Distance}$

$S=0.3\text{m}/25\text{m}$

$S=0.012$

Manning’s roughness coefficient, n for concrete—0.013

Manning Formula:

$v=\frac{1}{n}{R}^{2/3}{S}^{1/2}$

Simplified the formula:

$V_{25\text{m}}=1/n\times R^{0.67}\times S^{0.5}$

$V_{25\text{m}}=1/0.013\times {0.3}^{0.67}\times {0.012}^{0.5}$

$V_{25\text{m}}=3.76\text{m}/\text{s}$

Calculate the Flow Rate at the 25-meter Mark (Q_25m)

With the velocity, we can determine the flow rate:

$Q_{25\text{m}}=A\times V_{25\text{m}}$

$Q_{25\text{m}}=1.44{\text{m}}^2\times 3.76\text{m}/\text{s}$

$Q_{25\text{m}}=5.4{\text{m}}^3/\text{s}$

Account for Elevation Rise Over the Last 5 Meters

The water rises by 1.8 meters over the last 5 meters to reach Bandau Intake, so the total effective head $h_{\text{eff Bandau Intake}}$ just before the orifice is:

$h_{\text{eff Bandau Intake}}=h_{\text{eff 25m}}-1.8\text{m}$

$h_{\text{eff Bandau Intake}}=0\text{m}-1.8\text{m}$

$h_{\text{eff Bandau Intake}}=-1.8\text{m}$

There is a negative value, which means the water has to rise, and we’d expect a reduction in kinetic energy converting to potential energy to overcome the elevation.

Calculate the Velocity and Flow Rate

The velocity of water rise on slope can be calculated using Bernoulli’s equation:

$V=\sqrt{2\times g\times h_{eff}}$

We would then calculate the velocity, assuming there’s enough head to drive the flow:

$V_{\text{rise}}=\sqrt{2\times g\times h_{\text{eff Bandau intake}}}$

$V_{\text{rise}}=\sqrt{2\times 9.81\text{m}/{\text{s}}^{\text{2}}\times 1.8\text{m}}$

$V_{\text{rise}}=5.94\text{m}/\text{s}$

Due to the effective head is −1.8 m, therefore the Velocity rises is −5.94 m/s.

$Q_{\text{5}\text{m}}=A\times V$

$Q_{\text{5}\text{m}}=1.44{\text{m}}^2\times \left(-5.94\text{m}/\text{s}\right)$

$Q_{\text{5}\text{m}}=-8.55{\text{m}}^{\text{3}}/\text{s}$

Hence,

Total Flowrate ( $Q_{\text{25}\text{m}}+Q_{\text{5}\text{m}}$)

$Q_{\text{total}}=5.4{\text{m}}^3/\text{s}+\left(-8.55{\text{m}}^3/\text{s}\right)$

$Q_{\text{total}}=-3.15{\text{m}}^3/\text{s}$

Therefore, flow rate along the 30 m (25 m + 5 m rise) culvert is −3.15 m³/s.

The construction work could not be carried out according to the original design (refer to figure case 1) because the existing structure (Bandau Intake) is shallow, which is not the same as in the approved design drawing.

i) Construction according to the original design is impossible because the depth of the Bandau Intake does not exceed the depth of the Infiltration Gallery. If construction is to be carried out as well, Bandau Intake will need to be demolished first, and then a new Intake will need to be built.

ii) To ensure sufficient flow of raw water to the Bandau Intake, the appointed consultant needs to do a detailed survey including designing a booster pump system with a pipeline to the Intake Bandau.

iii) To ensure sufficient continuous water flow to the water treatment plant, the Government (Sabah State Water Department) needs to propose installing 2 units or more suitable booster pumps along with a pipe system that is directly connected to the existing pipe or Bandau intake raw water pipeline.