1.2 Water Supply Hydraulics & Pipe Networks

Water Supply Hydraulics & Pipe Networks

The delivery of water from treatment facilities to consumers relies on intricate pipe networks and pumping systems. A robust grasp of fluid mechanics and hydraulics is necessary for analyzing these networks, calculating energy losses, and sizing pumps.

Fundamental Hydraulic Principles

The Continuity Equation

For incompressible fluids like water, the mass flow rate remains constant throughout a pipe network. The continuity equation relates volumetric flow rate ($Q$) to flow velocity ($v$) and pipe cross-sectional area ($A$): $Q = A_1 v_1 = A_2 v_2$

The Energy Equation (Bernoulli)

The Energy Equation, an extension of Bernoulli's principle that accounts for friction and pump energy, states that the total energy at one point in a system equals the total energy at another point downstream, plus any energy added by pumps, minus any energy lost to friction: $\frac{p_1}{\gamma} + z_1 + \frac{v_1^2}{2g} + h_p = \frac{p_2}{\gamma} + z_2 + \frac{v_2^2}{2g} + h_L$ Where:

• $\frac{p}{\gamma}$ = Pressure head
• $z$ = Elevation head
• $\frac{v^2}{2g}$ = Velocity head
• $h_p$ = Head added by pump
• $h_L$ = Total head loss (major + minor)

Friction and Energy Losses

Major Losses: Hazen-Williams and Darcy-Weisbach

Major losses occur due to friction between the water and the inner walls of the pipe over a given length.

1. Darcy-Weisbach Equation: The theoretically rigorous method for calculating major head loss ($h_f$) is: $h_f = f \frac{L}{D} \frac{v^2}{2g}$ Where $f$ is the Darcy friction factor, which depends on the Reynolds number and relative pipe roughness (determined via the Moody chart or Colebrook equation).

2. Hazen-Williams Equation: In municipal water engineering, the empirical Hazen-Williams formula is overwhelmingly preferred due to its simplicity. In US Customary units: $h_f = \frac{10.67 L Q^{1.852}}{C^{1.852} D^{4.87}}$ Where:

• $h_f$ = friction head loss (ft)
• $L$ = length of pipe (ft)
• $Q$ = flow rate (cfs or gpm converted appropriately depending on the exact constant used)
• $C$ = Hazen-Williams roughness coefficient (dimensionless, higher = smoother)
• $D$ = pipe diameter (ft)

Minor Losses

Minor losses ($h_m$) are caused by turbulence generated at valves, fittings, bends, and changes in pipe diameter. They are calculated as a function of velocity head: $h_m = K \frac{v^2}{2g}$ Where $K$ is the minor loss coefficient specific to the fitting type. Total head loss is the sum of major and minor losses: $h_L = h_f + \sum h_m$.

Pumps and System Curves

To move water through a network, centrifugal pumps add energy ($h_p$) to overcome elevation differences and frictional losses.

System Head Curve

The system head curve represents the energy required to move various flow rates through the pipe system. It is defined as the static head (elevation difference plus any pressure difference) plus the dynamic head (friction and minor losses, which vary with $Q^2$): $H_{sys} = H_{static} + H_{dynamic}(Q)$

The operating point of a pump is the exact intersection of the pump's performance curve (provided by the manufacturer) and the system head curve.

Pump Affinity Laws

The affinity laws describe how changes in a pump's rotational speed ($N$) or impeller diameter ($D$) affect capacity ($Q$), head ($H$), and power ($P$). For changes in speed (with constant diameter):

1. Capacity (Flow): $ \frac{Q_1}{Q_2} = \frac{N_1}{N_2} $
2. Head: $ \frac{H_1}{H_2} = \left(\frac{N_1}{N_2}\right)^2 $
3. Power: $ \frac{P_1}{P_2} = \left(\frac{N_1}{N_2}\right)^3 $

Net Positive Suction Head and Cavitation

Cavitation occurs when the absolute pressure inside the pump falls below the vapor pressure of water, causing bubbles to form and rapidly collapse, damaging the impeller. To prevent cavitation, the Net Positive Suction Head Available (NPSHA) must be strictly greater than the Net Positive Suction Head Required (NPSHR), which is provided by the pump manufacturer.

$NPSHA = \frac{P_{atm}}{\gamma} + \frac{P_s}{\gamma} \pm z_s - \frac{P_{vapor}}{\gamma} - h_L$ Where:

• $\frac{P_{atm}}{\gamma}$ = Atmospheric pressure head
• $z_s$ = Suction static head (positive if source is above the pump, negative if lifting)
• $\frac{P_{vapor}}{\gamma}$ = Vapor pressure head of the liquid
• $h_L$ = Friction losses in the suction piping

Water Hammer

Water hammer is a pressure surge or wave caused when a fluid in motion is forced to stop or change direction suddenly (e.g., rapid valve closure). The magnitude of the pressure wave depends on the fluid's wave celerity (speed of sound in the fluid) and the velocity change. Surge tanks, pressure relief valves, and slow-closing valves are used to mitigate these transient forces.