Water Distribution Network: Multi-Reservoir Model

Hydraulic Junction Solver

Define the elevation and connecting pipe parameters for each reservoir. Add up to 6 reservoirs to solve complex single-junction systems.

Invalid parameters. Ensure all pipes have D > 0 and L > 0.

Solution

Junction Head H_j 0.00 m Converged in 0 iterations

Continuity Check (ΣQ) 0.00 L/s Error margin (must be ~ 0)

Hydraulic Profile Schematic Animated dashes indicate flow direction

Detailed Pipe Hydraulics

Pipe	Res Elev (m)	Diameter (mm)	Length (m)	Direction	Flow (L/s)	Velocity (m/s)	Head Loss (m)	Friction Slope (m/km)

Hydraulic Theory & Solver Methodology

1. The Principle of Continuity

At any pipe junction, according to the conservation of mass, the total flow entering the node must equal the total flow leaving it. In this N-reservoir system, the solver iteratively finds the unique junction head ($H_j$) that satisfies:

$$\sum_{i=1}^{N} Q_i = 0$$

2. Hazen-Williams Equation

Flow in each pipe depends on the head difference between the reservoir ($Z_i$) and the junction ($H_j$). Using the empirical Hazen-Williams formula (for metric units):

$$H_f = \frac{10.67 \cdot L \cdot Q^{1.852}}{C^{1.852} \cdot D^{4.87}}$$

Where $H_f = |Z_i - H_j|$. Solving for Q gives the flow rate for a given junction head.

3. Numerical Bisection Algorithm

Because the flow equation is non-linear ($Q \propto H_f^{0.54}$), a direct algebraic solution is impossible. This tool employs the Bisection Method. It sets the initial bounds for $H_j$ between the highest and lowest reservoir elevations. It guesses the midpoint, calculates the net flow $\Sigma Q$, and iteratively halves the search interval until $\Sigma Q \approx 0$ (within a $10^{-6}$ error margin). This method guarantees absolute convergence for any number of pipes.

References

• Hazen, A., & Williams, G. S. (1920). Hydraulic Tables: The Elements of Gagings and the Friction of Water Flowing in Pipes (3rd ed.). John Wiley & Sons.
• Mott, R. L., & Untener, J. A. (2014). Applied Fluid Mechanics (7th ed.). Pearson.
• Bhave, P. R. (1991). Analysis of Water Distribution Networks. Technomic Publishing Co.

Version 2.0 - Multi-Node Support

CivilSheets by Sothearith Seak