Estimating specific flow rates in drinking water distribution networks is essential to ensure optimal design, efficient operation, and sustainable infrastructure management. Several methods are used to determine these flow rates based on the demographic and urban characteristics of the service area. Among these methods, the most commonly adopted are the linear method, the area method, and the density method.

The linear method is based on the assumption that water demand is proportional to the length of pipes serving a given node. It is often used when detailed consumption data are not available. It is expressed as:

$q_{sp}=\frac{Q_p{}_{{}_{\text{domestic}}}}{{\displaystyle {\sum }_1^n{L}_i}}$(1)

$Q_{ri}=q_{sp}\times L_i$ (2)

$Q_{ni}=\frac{{\displaystyle {\sum }_{i=1}^{n=n}{Q}_{ri}}}{2}+{\displaystyle \sum Q_{\text{equipment}}}$(3)

where:

This method is not precise because we can have a long pipe that serves only a few subscribers; therefore, the resulting flow rate is very high. It is recommended that the equivalent length method be used instead of the geometric length. We associate a coefficient called the equivalent coefficient C with the geometric length [6] [7] .

The area method allocates demand based on the area served by a node. It is useful in residential areas where population density is relatively constant.

The method is based on dividing the total surface area into subsurface areas surrounded by nodes.

$q_{sp}=\frac{Q_p{}_{{}_{\text{domestic}}}}{{\displaystyle {\sum }_1^n{S}_i}}$ (4)

$Q_{ri}=q_{sp}\times S_i$(5)

where:

$S={\displaystyle {\sum }_1^n{S}_i}$

$Q_{ni}=\frac{{\displaystyle {\sum }_{i=1}^{n=n}{Q}_{ri}}}{2}+{\displaystyle \sum Q_{\text{equipment}}}$ (6)

Among the disadvantages of this method is that it does not take into account the density of inhabitants per surface area [6] .

The density method is based on the population or activity density in a given area. It is particularly used in urban and metropolitan areas where the population varies from one sector to another.

This is the most accurate method that provides the actual flow rate served by the section in question.

$q_{sp}=\frac{Q_p{}_{{}_{\text{domestic}}}}{{\displaystyle {\sum }_1^n{N}_{pi}}}$ (7)

The flow rate on the road is produced between the specific flow rate and the number of inhabitants served by the section considered. We can write:

$Q_{ri}=q_{sp}\times N_{pi}$ (8)

The flow rate at the node is given by the following relation:

$Q_{ni}=\frac{{\displaystyle {\sum }_{i=1}^{n=n}{Q}_{ri}}}{2}+{\displaystyle \sum Q_{\text{equipment}}}$ (9)

$N_{pi}$: Number of inhabitants served by the section considered (inhabitants).

${\displaystyle {\sum }_1^n{N}_{pi}}$: Number of inhabitants (inhab).

This method allows a more precise estimation of the specific flow rate by taking into account variations in population density and consumption habits [6] [8] .

The choice of method depends on the geographical and urban context of the area under study. The linear method is suitable for rural areas; the surface method is suitable for cities with homogeneous development, while the density method is preferred for dense urban centers. A combination of these approaches is sometimes necessary for a more reliable estimate and better management of water resources.

Although these methods remain useful, they face several challenges and obstacles, including:

These methods lack flexibility and adaptability to actual consumption dynamics.

In the face of these problems and limitations, we propose a new method that allows them to be considered and addressed.

This method takes into account the density of population or activities (commercial, industrial, etc.) in the area under study. It is more accurate than previous methods because it incorporates demographic or socioeconomic data to estimate water demand according to well-studied criteria. It is often used for urban areas where population density varies significantly.

The linear method consists of establishing the flow rate at nodes using the following procedure: First, the peak flow rate is calculated, then the total network length is calculated, which will be used to calculate the specific flow rate. Finally, the flow rate calculation phase at nodes using the linear method is carried out as follows:

To calculate the peak flow rate, the current and future population numbers must be estimated according to the projection horizon, the allocation (ℓ/d/inhab), the average daily flow rate must be calculated, the maximum daily flow rate must be calculated, the increase coefficients must be calculated, and finally, the peak flow rate is calculated:

$Q_p=Q_{avd}\cdot K_p$

where:

$K_p=K_h\cdot K_d$

K_h: hourly coefficient, which expresses the irregularity of the population. It is given by: K_h = α_max × β_max during the hours of the day.

α_max: coefficient that depends on the population's comfort level and the work schedule.

With: 1.2 < α_max < 1.4

β_max: coefficient that depends on the number of inhabitants. It varies between 0.2 and 1.15 for a population of less than 1000 to more than 50,000 inhabitants.

$Q_{sp}=\frac{Q_p}{L}$

where:

$Q_{sp}$: Specific flow rate.

$Q_p$: Peak flow rate.

$L$: Sum of all the lengths constituting the network.

The nodal flow rate is expressed by the following formula:

So, the flow rate of nodes:

$q_n=\frac{{\displaystyle \sum L_i}}{2}\times q_s$

where:

$q_n$: Node flow rate.

${\displaystyle \sum L_i}$: Sum of the lengths of the adjacent segments of this node.

$q_{sp}$: Specific flow rate [11] [12] .

These results are summarized in “ Table 1 ” and “ Table 2 ”, respectively. These are the nodes results table and the arcs results table, after performing hydraulic simulation on a water distribution network model using EPANET software in both the linear method and the collective buildings method. These two tables are essential elements resulting from hydraulic simulation.

The node results in “ Table 1 ” provide essential information on the hydraulic conditions at each network junction point, including pressure, water demand, and elevation. It allows for the evaluation of the system’s performance in terms of user service and compliance with regulatory pressure thresholds. These data are crucial for diagnosing hydraulic imbalances and guiding network improvements.

This table provides important information about the flow, including: (flow rate, speed, diameter, head loss)

The arc results “ Table 2 ” report the flow characteristics in each pipe segment, such as discharge, water velocity, and head loss. They allow for the analysis of the efficiency of water transport between nodes and the identification of critical sections likely to cause malfunctions. These results are used to optimize pipeline design and operation.

The objective of this example is to explain the methodology applied to calculate flow rates at nodes and to compare the results obtained using this method with those obtained using the conventional method (linear method) used to calculate the Massinissa city network in Algeria.

The methodology consists of:

1) Dividing the study area into sectors based on the type and number of floors of the buildings.

2) Calculating the flow rate for each sector of the study area using the formula above, taking into account the number of dwellings per floor and the associated heights.

3) Calculate the total flow rate by summing the flow rates for each sector.

4) Calculating the total number of dwellings in the study area.

5) Calculating the specific flow rate for the study area in ℓ/s/apartment.

For the application case, our study area is divided into four sectors as follows:

Sector 4th floor with 2, 3, and 5 apartments on each floor.

Sector 5th floor with 3, 4, and 6 apartments on each floor.

Sector 9th floor with four apartments on each floor.

Sector 11th floor with four apartments on each floor.

Therefore, the calculation results are summarized in tables in the order indicated in the methodology.

While the use of simultaneity coefficients (Ks) and demand estimation based on sanitary fixtures and equipment, as well as the sanitary equipment present in apartments, are common in the design of building-scale plumbing systems, the novelty of our approach lies in the extension of these principles to urban water distribution modeling.

Specifically, the method innovatively sectorizes the study area based on building type and height, incorporates localized apartment data based on standards, and applies these values to node-level simulations within EPANET. By accounting for vertical density and occupancy patterns, this method allows for a more accurate and realistic spatial assignment of demand across the network.

1) Calculation of flow rates at nodes

The flow rate for each node is calculated using the formula:

$q_n=q_{sp}\cdot N_a$

$q_n$: flow rate at the node in (ℓ/s)

$q_{sp}$: specific flow rate in (ℓ/s/apart)

$N_a$: number of apartments connected to this node ( Table 4 ).

2) Equipment flow rate

For equipment flow rates, we have classified this equipment into two types as follows:

Equipment attached to buildings (commercial and service).

Equipment not attached to buildings (administration, primary schools, and middle school).

For equipment attached to buildings, the corresponding flow rates are already calculated and supported in the nodes that serve the buildings. See “ Tables 5-8 ”.

For equipment not attached to buildings, their flow rates are calculated according to the characteristics of each type.

This part of the work consists of analyzing and interpreting the results obtained by each method. The principle of analysis and interpretation will be based primarily and mainly on:

1) Analysis of the flowchart of each method

Flowchart of the Linear Method ( Figure 5 )

Flowchart of the collective buildings ( Figure 6 )

2) Analysis of the flow values at the nodes obtained by each method

By comparing and analyzing “ Table 9 ” and “ Table 10 ” concerning the flow rates at the nodes in the two methods, we observe the following:

For the linear method, all nodes have a base water demand, while in the collective buildings method, there are 35 nodes that do not have a base water demand and are solely connection junctions.

From this, we can conclude the following:

1) For the collective buildings method

After locating the nodes that do not have basic flow demand values, as shown in the tables above, we note that these nodes are considered sharing or change-of-direction points and have no other role to play in the distribution network, which is why the basic demand value is zero.

In this method:

The flow rate is assigned only to nodes where a building (or group of buildings) is actually connected.

It more realistically represents the location of demand, especially in densely populated areas.

If a node is not connected to a building, then it has no demand of its own and therefore no assigned flow rate.

As a result, only nodes connected to buildings have a non-zero nodal flow rate.

2) For the linear method

By comparing “ Table 9 ” and “ Table 10 ”, we note that in the collective buildings method, some nodes, which have base demand values in the linear method, do not take these values into account. It is worth noting that these nodes alone have a total base demand flow of 58.57 ℓ/s, out of a total of 78.64 ℓ/s, representing 74.5% of the total demand.

By comparing the two methods, we note a crucial point: the profitability of distributing baseline demand across the entire network.

In the linear method, the distribution does not take into account the actual location of demand points. Therefore, there will be a sudden increase in baseline demand in areas where it is not necessarily justified.

This highlights the main weakness of the linear method: its tendency to distribute demand proportionally to pipe length without taking into account the reality on the ground (population density, building availability, etc.). Conversely, the collective buildings method, which is based on more realistic data, allows for better distribution of demand, which can lead to a more economical design that is more accurate for the actual needs of the network.

1) Velocity analysis

We observe from methods, the linear method and the construction method, that the flow velocity in water distribution network pipes ranges from 0.50 m/s to 1.50 m/s, which is within the acceptable range.

2) Analysis of flows

Since flows are closely related to peak flows and flows at nodes, the difference between the flows of the two methods will necessarily be clear and logical.

We note that the network in the study area is well-dimensioned in relation to the flows carried by the different sections of the network, whether calculated by the linear method or the collective buildings method.

As for the diameters, they are proportional to the flow quantity, as evidenced by the fact that the velocity values are proportional to the optimal range.