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Fluid Mechanics

9.3 Pipe Network Analysis

3 min readLast Updated on July 19, 2024

Pipe networks are complex systems that rely on conservation principles to function properly. Understanding mass and energy conservation helps engineers analyze flow rates, pressure changes, and energy losses throughout interconnected pipes.

The Hardy Cross method is a powerful tool for solving pipe network problems. By iteratively adjusting flow rates, engineers can determine the actual flow distribution, calculate pressure drops, and optimize pumping power to compensate for head losses in the system.

Pipe Network Analysis

Conservation principles for pipe networks

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  • Conservation of mass principle maintains that the flow rate entering a junction must equal the flow rate leaving the junction (Qin=Qout\sum Q_{in} = \sum Q_{out}) ensuring continuity of flow throughout the network (pipe junctions, splits, and merges)
  • Conservation of energy principle applies Bernoulli's equation to pipe networks accounting for pressure, velocity, elevation, and head loss due to friction along the pipes (p1ρg+v122g+z1=p2ρg+v222g+z2+hL\frac{p_1}{\rho g} + \frac{v_1^2}{2g} + z_1 = \frac{p_2}{\rho g} + \frac{v_2^2}{2g} + z_2 + h_L)
  • Continuity equation for incompressible flow states that the flow rate remains constant along a pipe with varying cross-sectional areas (Q=v1A1=v2A2Q = v_1 A_1 = v_2 A_2) enabling calculation of velocity changes (pipe expansions, contractions)
  • Energy loss due to friction in pipes can be determined using the Darcy-Weisbach equation (hL=fLDv22gh_L = f \frac{L}{D} \frac{v^2}{2g}) for turbulent flow or the Hazen-Williams equation (hL=10.67LC1.852D4.87Q1.852h_L = \frac{10.67 L}{C^{1.852} D^{4.87}} Q^{1.852}) for water flow in pipes

Hardy Cross method for flow analysis

  • Hardy Cross method provides an iterative approach to solve pipe network problems by assuming an initial flow distribution and correcting the flow rates based on the conservation principles
  • Steps in the Hardy Cross method:
    1. Assign an initial flow distribution (clockwise or counterclockwise) to the network
    2. Calculate the head loss in each pipe using the Darcy-Weisbach or Hazen-Williams equation based on the assumed flow rates
    3. Calculate the flow correction factor for each loop using the formula ΔQ=loophLloophLQ\Delta Q = \frac{\sum_{loop} h_L}{\sum_{loop} \frac{h_L}{Q}} to adjust the flow rates
    4. Update the flow rates in each pipe by adding the correction factor: Qnew=Qold+ΔQQ_{new} = Q_{old} + \Delta Q
    5. Repeat steps 2-4 until the flow correction factors are within an acceptable tolerance indicating convergence to the actual flow distribution
  • Pressure drops in pipe networks can be calculated using the head loss equations and converting head loss to pressure drop: Δp=ρghL\Delta p = \rho g h_L (water distribution systems, industrial piping)

Pumping power for head loss compensation

  • Pumping power represents the power required to overcome head losses and maintain desired flow rates in a pipe network calculated using the formula P=ρgQHP = \rho g Q H where PP is the pumping power (W), ρ\rho is the fluid density (kg/m³), gg is the acceleration due to gravity (m/s²), QQ is the flow rate (m³/s), and HH is the total head (m)
  • Total head consists of the sum of static head (elevation difference between the pump and the discharge point) and head losses (friction losses in pipes and minor losses in fittings and valves)
  • Pumping power calculations help in selecting appropriate pump sizes and determining energy consumption in pipe networks (water supply systems, irrigation networks, industrial processes)

Optimization of pipe network design

  • Pipe sizing plays a crucial role in optimizing pipe networks as larger pipe diameters result in lower head losses but higher material costs, requiring a balance between head loss and material costs which can be determined using economic analysis or optimization techniques (cost-benefit analysis, life-cycle costing)
  • Network configuration, the arrangement of pipes, junctions, and pumps, affects the overall head loss and pumping requirements where parallel pipes can reduce head loss compared to a single pipe while series pipes increase the total head loss (parallel and series pipe configurations)
  • Optimization techniques such as linear programming, genetic algorithms, and particle swarm optimization can be employed to find the optimal pipe sizes and network configurations
  • Objective functions for optimization may include minimizing total head loss, pumping costs (energy consumption), material costs (pipe sizes), or a combination of these objectives depending on the specific requirements and constraints of the pipe network (water distribution networks, gas pipelines, district heating systems)
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© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.

© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.