Pipe network analysis (e.g., municipal water distribution): which fundamental relations must simultaneously be satisfied at steady state?

Difficulty: Medium

Correct Answer: All the above.

Explanation:


Introduction / Context:
Design and analysis of looped pipe networks require solving for unknown flows and nodal heads. Classical methods (Hardy Cross, linear theory) and modern solvers enforce mass and energy conservation with appropriate head-loss laws for each pipe segment.


Given Data / Assumptions:

  • Steady-state incompressible flow.
  • Known demands or supplies at nodes.
  • Empirical/analytical head-loss relation (e.g., Darcy–Weisbach or Hazen–Williams).


Concept / Approach:

Two conservation laws govern the solution: continuity at nodes and energy around loops. Each pipe requires a constitutive relation linking discharge to head loss. Together, these equations form a solvable system for flows and heads.


Step-by-Step Solution:

Continuity: ΣQ_in − ΣQ_out = 0 at every junction.Energy: ΣΔh around each independent loop = 0 (accounting for pumps/minor losses).Head loss: Δh_f = f * (L/D) * (V^2/(2g)) (Darcy–Weisbach) or alternative relation.Iteratively adjust flows until both continuity and loop energy balances are satisfied.


Verification / Alternative check:

Compute residuals of nodal continuity and loop energy; convergence to near zero confirms a physically consistent solution.


Why Other Options Are Wrong:

Each of (a), (b), and (c) is necessary; omitting any yields an under-defined or inconsistent system. Hence “All the above” is correct.


Common Pitfalls:

Using inconsistent units; neglecting minor losses or pump curves; applying an inappropriate head-loss formula outside its range.


Final Answer:

All the above.

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